Theory of constructive semigroups with apartness -- foundations, development and practice
Melanija Mitrovic, Mahouton Norbert Hounkonnou, Marian Alexandru Baroni
aa r X i v : . [ m a t h . G M ] A ug THEORY OF CONSTRUCTIVE SEMIGROUPS WITH APARTNESS- FOUNDATIONS, DEVELOPMENT AND PRACTICE
Melanija Mitrovi´c , Mahouton Norbert Hounkonnou and Marian Alexandru Baroni Faculty of Mechanical Engineering, University of Niˇs, Serbia International Chair in Mathematical Physics and Applications (ICMPA-UNESCOChair), University of Abomey-Calavi, 072 B.P. 50 Cotonou, Republic of Benin “Dunarea de Jos” University of Galati, Romania August 26, 2020 “Semigroups aren’t a barren, sterile flower on the tree of algebra, they are a natural algebraic approach to some of themost fundamental concepts of algebra (and mathematics in general), this is why they have been in existence for more thenhalf a century, and this is why they are here to stay.” Boris M. Schein, [47]
Abstract
This paper has several purposes. We present through a critical review the resultsfrom already published papers on the constructive semigroup theory, and contributeto its further development by giving solutions to open problems. We also drawattention to its possible applications in other (constructive) mathematics disciplines,in computer science, social sciences, economics, etc. Another important goal of thispaper is to provide a clear, understandable picture of constructive semigroups withapartness in Bishop’s style both to (classical) algebraists and the ones who applyalgebraic knowledge.
Keywords.
Semigroup with apartness, set with apartness, co-quasiorder, co-equivalence,co-congruence.
MSC (2010) 03F65, 20M99
A general answer to the question what constructive mathematics is could be formulated asfollows: it is mathematics which can be implemented on a computer. There are two mainways of developing mathematics constructively. The first one uses classical traditional logicwithin a strict algorithmic framework. The second way is to replace classical logic with intuitionistic logic .Throughout this paper constructive mathematics is understood as mathematics per-formed in the context of intuitionistic logic, that is, without the law of excluded middle(
LEM ). There are two main characteristics for a constructivist trend. The notion of truth
1s not taken as primitive, and existence means constructibility. From the classical mathe-matics (
CLASS ) point of view, mathematics consists of a preexisting mathematical truth.From a constructive viewpoint, the judgement ϕ is true means that there is a proof of ϕ .“What constitutes a proof is a social construct, an agreement among people as to what is avalid argument. The rules of logic codify a set of principles of reasoning that may be usedin a valid proof. Constructive (intuitionistic) logic codifies the principles of mathematicalreasoning as it is actually practiced,” [23]. In constructive mathematics, the status of exis-tence statement is much stronger than in CLASS . The classical interpretation is that anobject exists if its non-existence is contradictory. In constructive mathematics when theexistence of an object is proved, the proof also demonstrates how to find it. Thus, followingfurther [23], the constructive logic can be described as logic of people matter, as distinctfrom the classical logic, which may be described as the logic of the mind of God. Oneof the main features of constructive mathematics is that the concepts that are equivalentin the presence of
LEM , need not be equivalent any more. For example, we distinguishnonempty and inhabited sets, several types of inequalities, two complements of a given set,etc.There is no doubt about deep connections between constructive mathematics and com-puter science. Moreover, “if programming is understood not as the writing of instructionsfor this or that computing machine but as the design of methods of computation that is thecomputer’s duty to execute, then it no longer seems possible to distinguish the disciplineof programming from constructive mathematics”, [33].Constructive mathematics is not a unique notion. Various forms of constructivismhave been developed over time. The principle trends include the following varieties:
INT - Brouwer’s intuitionistic mathematics,
RUSS - the constructive recursive mathematicsof the Russian school of Markov,
BISH - Bishop’s constructive mathematics. Every formhas intuitionistic logic at its core. Different schools have different additional principles oraxioms given by the particular approach to constructivism. For example, the notion ofan algorithm or a finite routine is taken as primitive in
INT and
BISH , while
RUSS operates with a fixed programming language and an algorithm is a sequence of symbols inthat language. We have to emphasize that Errett Bishop - style constructive mathematics,
BISH , forms the framework for our work.
BISH enables one to interpret the resultsboth in classical mathematics and in other varieties of constructivism.
BISH originatedin 1967 with the publication of the book
Foundations of Constructive Mathematics , [4],and with its second, much revised edition in 1985, [5]. There has been a steady stream ofpublications contributing to Bishop’s programme since 1967. A ten-year long systematicresearch of computable topology, using apartness as the fundamental notion, resulted in thefirst book, [12], on topology within
BISH framework.
Modern algebra , as is noticed in[10], “contrary to Bishop’s expectations, also proved amenable to natural, thoroughgoing,constructive treatment”.Working within the classical theory of semigroups several years ago we, [36], decidedto change the classical background with the intuitionistic one. This meant, among otherthings, that the perfect safety of the classical theory with developed notions, notations andmethodologies was left behind. Instead, we embarked on an adventure into exploring an2lgebraically new area (even without clearly stated notions and notations) of constructivesemigroups with apartness . What we had “in hand” at that moment was the experienceand knowledge coming from the classical semigroup theory, other constructive mathematicsdisciplines such as, for example, constructive analysis, and, especially, from constructivetopology, as well as constructive theories of groups and rings with tight apartness andcomputer science. For classical algebraists who, like us, wonder “on the odd day” whatconstructive algebra is all about, and who want to find out what it feels like doing it,they will understand soon that constructive algebra is more complicated than classicalalgebra in various ways: algebraic structures as a rule do not carry a decidable equalityrelation (this difficulty is partly met by the introduction of a strong inequality relation,the so-called apartness relation); there is (sometime) the awkward abundance of all kindsof substructures, and hence of quotient structures, [51].
The theory of semigroup with apartness is a new approach to semigroup theory and nota new class of semigroups. This paper has several purposes: to present through a criticalreview results from our already published papers, [16], [17], [38], on the constructive pointof view on semigroup theory, to contribute to its further development giving solutions tothe problems posted (in the open, or, somehow hidden way) within the scope of thosepapers, and to lay the foundation for further works.The order theory provides one of the most basic tools of semigroup theory within
CLASS . In particular, the structure of semigroups is usually most clearly revealed throughthe analysis of the behaviour of their appropriate orders. The most basic concept leadsto the quasiorders , reflexive and transitive relations with the fundamental concepts beingintroduced whenever possible in their natural properties. Going through [16], [17], [38], wecan conclude that one of the main objectives of those papers is to develop an appropriateconstructive order theory for semigroups with apartness. We outline some of the basicconcepts of semigroups with apartness, as special subsets on the one hand, and orders onthe other. The strongly irreflexive and co-transitive relations are building blocks of theconstructive order theory we develop. With a primitive notion of ’set with apartness’ ourmain intention was to connect all relations defined on such a set. This is done by requiringthem to be a part (subset) of an apartness. Such a relation is clearly strongly irreflexive.If, in addition, it is co-transitive, then it is called co-quasiorder .In algebra within
CLASS , the formulation of homomorphic images (together withsubstructures and direct products) is one of the principal tools used to manipulate algebraicstructures. In the study of homomorphic images of an algebraic structure, a lot of helpcomes from the notion of a quotient structure, which captures all homomorphic images, atleast up to isomorphism. On the other hand, the homomorphism is the concept which goeshand in hand with congruences. The relationship between quotients, homomorphisms andcongruences is described by the celebrated isomorphism theorems , which are a general andimportant foundational part of abstract and universal algebras. The quotient structuresare not part of
BISH . The quotient structure does not, in general, have a natural apartnessrelation. So, the Quotient Structure Problem ( QSP ) is one of the very first problem whichhas to be considered for any structure with apartness. The solutions of
QSP problem forsets and semigroups with apartness were for the first time given in [16].
Co-equivalences ,3ymmetric co-quasiorders, and equivalences which can be associated to them play the mainroles. As an example that a single concept of classical mathematics may split into two ormore distinct concepts when working constructively we have logical ¬ Y and apartness ∼ Y complement of a given subset Y of a set or semigroup with apartness. The key for thesolution of the QSP for a set and semigroup with apartness is given by the next theorem(Theorem 2.3, [16]).
Theorem 1.1. If κ is a co-equivalence on S , then the relation ∼ κ is an equivalence on S ,and κ defines apartness on S/ ∼ κ . Theorem 1.1 is the key ingredient to formulate and prove the apartness isomorphismtheorem for a set with apartness (see [16], Theorem 2.5). Based on these results, theapartness isomorphism theorem for a semigroup with apartness ([16], Theorem 3.4) isformulated and proved as well. The just mentioned results are significantly improvedin [38], where, among other things, it is proved that the two complements, logical andapartness, coincide for a co-quasiorder τ defined on a set or semigroup with apartness, i.e.we have ∼ τ = ¬ τ (see Proposition 2.3, Theorem 2.4 from [38]). Remark 1.
In [37], an overview to the development of isomorphism theorems in certainalgebraic structures - from classical to constructive - is given.
It is well known that within
CLASS a number of subsets of a semigroup enjoy spe-cial properties relative to multiplication, for example, completely isolated subset, convexsubset, subsemigroup, ideal. On the other hand, relations defined on a semigroup can bedistinguished one from another according to the behaviour of their related elements tomultiplication. From that point of view, positive quasiorders are of special interest. Goingthrough literature with the theory of semigroups as the main topic, one can see that thereis almost no method of studying semigroups without a certain type of positive quasiordersinvolved. It is often the case that results on connections between positive quasiorders andsubsets defined above showed them fruitful as well. Partly inspired by classical results,in [17] we consider complement positive co-quasiorders , i.e. constructive counterparts ofpositive quasiorders, and their connections with special subsets of semigroups with apart-ness. Inspired by the existing notion from constructive analysis and topology, we use thecomplements (both of them) for the classification of subsets of a given set with apartness.
Strongly detachable subsets , i.e. those subsets for which we can decide whether an element x from that set belongs to the subset in question or to its apartness complement, play asignificiant role within the scope of [17]. In Lemma 3.2, we prove that for any co-quasiorder τ , defined on a set with apartness, its left and right τ -classes of any element are stronglydetachable subsets. The main result of this paper, Theorem 4.1, gives the descriptionof a complement positive co-quasiorder, defined on a semigroup with apartness, via thebehaviour of its left and right classes and their connections with special subsets.Apart from the above issues, an addition problem is the so-called constant domainaxiom : the folklore type of axiom in CLASS algebra ϕ ∨ ∀ x ψ ( x ) ↔ ∀ x ( ϕ ∨ ψ ( x )) , ϕ ∨ ∀ x ψ ( x ) → ∀ x ( ϕ ∨ ψ ( x ))can be a source of problems when doing algebra constructively. We choose intuitionisticlogic of constant domains CD to be the background of [17]. Recall that intermediate logic,such as, for example CD , the logic that is stronger than intuitionistic logic but weakerthan classical one, can be constructed by adding one or more axioms to intuitionistic logic.There is a continuum of such logics.The presence of apartness implies the appearance of different types of substructuresconnected to it. We deal with strongly detachable subsets in [17]. In [38] we mentionedtwo more: detachable and quasi-detachable subsets. In Proposition 2.1 we show that astrongly detachable subset is detachable and quasi-detachable. Even more, from [38] theapartness and logical complements coincide for strongly detachable and quasi-detachablesubsets.Going through [17] and [38] it can be noticed that we can face several problems arisingfrom their scope. For example,- The relations between detachable, strongly detachable and quasi detachable subsetsare only partially described in [38], Proposition 2.1. A complete description of their rela-tionship remains an open problem.- Are the results of [17] valid in intuitionistic logic if we work with quasi-detachablesubsets instead of strongly detachable ones? Which of the presented results or their form(s)are valid for the intuitionistic background, if any?To conclude, the theory of semigroup with apartness, its background and motivations,further development and its possible applications as well as the critical answers to a numberof questions including those mentioned above will be the main topics throughout this paper.The paper is organized in the following way. In our work on constructive semigroupswith apartness, as it is pointed out above, we have faced an algebraically completely newarea. The background and motivation coming from the classical semigroup theory, otherconstructive mathematics disciplines and computer science are the content of Section2 . Some results on classical semigroups which can partly be seen as an inspiration forthe constructive ones are also discussed here. In
Section 3 , the main one, we are goingto give a critical review of some of the published results on sets and semigroups withapartness as well as the solutions to some of the open problems on sets and semigroupswith apartness. One of the main results, Theorem 3.1, gives a complete description ofthe relationships between distinguished subsets of a set with apartness, which, in turn,justifies the constructive order theory we develop with those subsets with the main role inthat framework. By Proposition 3.2, if any left/right-class of a co-quasiorder defined ona set with apartness is a (strongly) detachable subset then
LPO holds. This shows thatTheorem 4.1 on the complement positive co-quasiorder (and Lemma 3.2 important for itsproof) from [17] cannot be proved in
BISH without the logic of constant domains CD .Within intuitionistic logic, we can prove its weaker version, Theorem 3.7, which is anotherimportant result of this section. As for QSP , for sets and semigroups with apartness, weachieve a little progress in that direction. Theorem 3.2, the key theorem for the
QSP ’s5olution, generalizes the similar ones from [16], [38]. In addition, as a generalization of thefirst apartness isomorphism theorem, the new theorem, Theorem 3.4, the second apartnessisomorphism theorem for sets with apartness, is formulated and proved. Finally, in
Section4 , examples of some already existing applications as well as future possible realizations ofthe ideas presented in the previous section are given.More background on constructive mathematics can be found in [3], [4], [12], [51]. Thestandard reference for constructive algebra is [35]. For the classical case see [36], [41].Examples of applications of these theoretical concepts can be found in [2], [11], [15] [22],[40].
Starting our work on constructive semigroups with apartness, as pointed out above, wefaced an algebraically completely new area. What we had in “hand” at that momentwere the experience and knowledge coming from the classical semigroup theory, otherconstructive mathematics disciplines, and computer science. “I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads ofarithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmeticalformula: 2 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We aremere operatives, empirics, and egotists until we learn to think in letters instead of figures .”Oliver Wendell Holmes:
The Autocrat of the Breakfast Table
A very short account of abstract algebra and its development will be given here. Overthe course of the 19th century, algebra made a transition from a subject concerned entirelywith the solution of mostly polynomial equations to a discipline that deals with generalstructures within mathematics. The term abstract algebra as a name for this area appearedin the early 20th century. “In studying abstract algebra, a so called axiomatic approachis taken; that is, we take a collection of objects S and assume some rules about theirstructure. These rules are called axioms. Using the axioms for S , we wish to derive otherinformation about S by using logical arguments. We require that our axioms be consistent;that is, they should not contradict one another. We also demand that there not be to manyaxioms. If a system of axioms is too restrictive, there will be few examples of the algebraicstructure,” [30].An algebraic structure can be, informally, described as a set of some elements ofobjects with some (not necessarily, but often, binary) operations for combining them. A set is considered as a primitive notion which one does not define. We will take the intuitiveapproach that a set is some given collection of objects, called elements or members of the6et. The cartesian product of a set S with itself, S × S , is of special importance. A subset ρ of S × S , or, equivalently, a property applicable to elements of S × S , is called a binaryrelation on S . The ordered pair ( S, ρ ) is a particular relational structure . In general, thereare many properties (for example: reflexivity, symmetry, transitivity) that binary relationsmay satisfy on a given set. As usual, for a relation ρ on S , aρ = { x ∈ S : ( a, x ) ∈ ρ } , and ρa = { x ∈ S : ( x, a ) ∈ ρ } are the left and the right ρ -class of the element a ∈ S respectively.The concept of an equivalence , i.e. reflexive, symmetric and transitive relation, is anextremely important one and plays a central role in mathematics. If ε is an equivalence ona set S , then S/ε = { xε : x ∈ S } is called the quotient set of S by ε . Classifying objectsaccording to some property is a frequent procedure in many fields. Grouping elementsin “a company” so that elements in each group are of the same prescribed property asperformed by equivalence relations, and the classification gives the corresponding quotientsets. Thus, abstract algebra can show us how to identify objects with the same propertiesproperly - we have to switch to a quotient structure (technique applicable, for example, toabstract data type theory).Some fundamental concepts in abstract algebra are: set and operation(s) defined onthat set; certain algebraic laws that all elements of a structure can respect, such as, forexample, associativity, commutativity; some elements with special behaviour in connectionwith operation(s): idempotent elements, identity element, inverse elements, ... Combiningthe above concepts gives some of the most important structures in mathematics: groups,rings, semigroups, ... Centred around an algebraic structure are notions of: substructure,homomorphism, isomorphism, congruence, quotient structure. A mapping between twoalgebraic structures of the same type, that preserves the operation(s) or is compatible withthe operation(s) of the structures is called homomorphism . Homomorphisms are essentialto the study of any class of algebraic objects. An equivalence relation ρ on an algebraicstructure S (such as a group, a ring, or a semigroup) that is compatible with the structureis called a congruence . Within CLASS the quotient set
S/ρ becomes the structure ofthe same type in a natural way. The relationship between quotients, homomorphismsand congruences is described by the celebrated isomorphism theorems . Isomorphismtheorems are a general and important foundational part of abstract and universal algebra.“Algebra is beautiful. It is so beautiful that many people forget that algebra can be veryuseful as well,” [32]. Abstract algebra is the highest level of abstraction. Understandingit means, among other things, that one can think more clearly, more efficiently. With thedevelopment of computing in the last several decades, applications that involve algebraicstructures have become increasingly important. To mention a few, lot of data structuresform monoids (semigroups with the identity element); algebraic properties are importantfor parallel execution of programs - for example, combining a list of items with somebinary operators can be easily parallelized if that operator is associative (commutativity isoften required as well). Examples of applications given above lead to semigroups . In fact,following [31], (free) semigroups are the first mathematical objects every human being hasto deal with - even before attending school. 7 .1.1 More about the theory of semigroups A semigroup is an algebraic structure consisting of a set with an associative binary op-eration defined on it. In the history of mathematics, the algebraic theory of semigroups is a relative newcomer, with the theory proper developing only in the second half of thetwentieth century. Historically, it can be viewed as an algebraic abstraction of the prop-erties of the composition of transformations on a set. But, there is no doubt about it, themain sources came from group and ring theories. However, semigroups are not a directgeneralization of group theory as well as ring theory. Let us remember: congruences ongroups are uniquely determined by their normal subgroups, and, on the other hand, thereis a bijection between congruences and the ideals of rings. The study of congruences onsemigroups is more complicated - no such device is available. One must study congruencesas such. Thus, semigroups do not much resemble groups and rings. In fact, semigroupsdo not much resemble any other algebraic structure. Nowadays, semigroup theory is anenormously broad topic and has advanced on a very broad front. Following [34], “a hugevariety of structures studied by mathematicians are sets endowed with associative binaryoperation.” Even more, it appears that “semigroup theory provides a convenient generalframework for unifying and clarifying a number of topics in fields that are seen, at firstsight, unrelated”, [31].The capability and flexibility of semigroups from the point of view of modelling andproblem-solving in extremely diverse situations have been already pointed out, and inter-esting new algebraic ideas arise with binary applications and connections to other areas ofmathematics and sciences. Let us start our short journey through the applications of semi-groups with the connections to the algebra of relations. The theory of semigroups is oneof the main algebraic tools used in the theory of automata as well as the theory of formallanguages. According to some authors, the role of the theory of semigroups for theoreticalcomputer science is compared with the one which the philosophy has with the respectto science in general. Some investigations on transformation semigroups of synchronizingautomata show up interesting implications for various applications for robotics, or moreprecisely, robotic manipulation. On the other hand, areas such as biology, biochemistry,sociology also make use of semigroups. For example, semigroups can be used in biologyto describe certain aspects in the crossing of organisms, in genetics, and in considerationof metabolisms. Following [7], [32], the sociology includes the study of human interactivebehaviour in group situations, in particular in underlying structures of societies. The studyof such relations can be elegantly formulated in the language of semigroups. The book [8]is written for social scientists with the main aim to help readers to apply “interesting andpowerful concepts” of semigroup theory to their own fields of expertise. However, the listof applications given above does not purport to mention all of the existing applications ofsemigroup theory. As it is pointed out in [34], it is often the case that “most applicationsmake minimal use of the reach of the (classical) algebraic theory of semigroups.” Thereis need for study of some more structures of semigroups which can find applications indifferent areas, [43]. This can bring very pretty mathematics to illustrate the interplaybetween certain scientific areas and semigroup-theoretic techniques. This type of research8an be a topic on its own for certain types of papers.In what follows some known results from the classical semigroup theory useful for ourdevelopment will be presented. A semigroup ( S, · ) is a set S together with an associative binary operation · (A) ( ∀ a, b, c ∈ S ) [( a · b ) · c = a · ( b · c )].Where the nature of the multiplications is clear from the context, it is written S ratherthan ( S, · ). Frequently, xy is written rather than x · y .Various approaches have been developed over the years to construct frameworks forunderstanding the structure of semigroups. The fundamental concepts of semigroup theoryelaborated by Suschekewitsch, Rees, Green, Clifford and other pioneers include as one ofthe main tools, Green’s quasiorders (and equivalences generated by them), defined by themultiplication of semigroups and in terms of special subsemigroups. The notion of an orderplays an important role throughout mathematics as well as in some adjacent disciplinessuch as logic and computer science. Order theory provides one of the most basic toolsof semigroup theory as well. In particular, the structure of semigroups is usually mostclearly revealed through the analysis of the behaviour of their appropriate orders. Apure order theory is concerned with a single undefined binary relation ρ . This relation isassumed to have certain properties (such as, for example, reflexivity, transitivity, symmetry,antisymmetry), the most basic of which leads to the concept of quasiorder . A quasiorderplays a central role throughout this short exposition with the fundamental concepts beingintroduced whenever possible in their natural properties. Distinguishing subsets
A number of subsets of a semigroup enjoy special properties relative to the multiplica-tion. A subset T of a semigroup S is: • completely isolated if ab ∈ T implies a ∈ T or b ∈ T for any a, b ∈ S , • convex if ab ∈ T implies both a, b ∈ T for any a, b ∈ S , • subsemigroup if for any a, b ∈ T we have ab ∈ T , • ideal if for any a ∈ T and s ∈ S we have as, sa ∈ T .A subsemigroup T of S which is convex (resp. completely isolated) as a subset is called a convex (resp. completely isolated ) subsemigroup . In an analogous way, we define a complex(completely isolated) ideal of S . Some of their existing properties are listed in the lemmabelow. Lemma 2.1.
Let S be a semigroup. Then: (i) An ideal I of S is completely isolated if and only if ¬ I = S \ I is either a subsemigroupof S or is empty. An nonempty subset F of S is convex if and only if ¬ F = S \ F is either a completelyisolated ideal or is empty. Within
CLASS semigroups can historically be viewed as an algebraic abstraction ofthe transformations on a set. Of great importance is the role of the subsemigroups givenabove in describing the structure of transformation semigroups. We refer the reader to [19],[48] for more details about definitions, properties and applications of such subsemigroups.Describing a semigroup and its structure is a formidable task. There are many differenttechniques developed for that purpose.
Semilattice decomposition of semigroups is one ofthe methods with general applications. For more information on semilattice decompositionof semigroups see [36], [41]. It is shown in [41] that this method leads to the study ofcompletely isolated ideals and convex subsemigroups.
Quasiorders
By definition, a binary relation ρ of set S is a subset of S × S . To describe the relationdefined on a semigroup S , we have to say which order pairs belong to ρ . In other words,for any a ∈ S , we have to know the following subsets of S : aρ = { x ∈ S | ( a, x ) ∈ ρ } , ρa = { x ∈ S | ( x, a ) ∈ ρ } ,called the left and right ρ -class of an element a . That is how we connect a study of binaryrelations defined on a given set with its subsets.The relations defined on a semigroup S are distinguished one from another accordingto the behaviour of their related elements to the multiplication. A relation ρ defined on asemigroup S is • positive if ( a, ab ) , ( a, ba ) ∈ ρ , for any a, b ∈ S , • with common multiply property , or, also called for short, with cm-property if ( a, c ) , ( b, c ) ∈ ρ implies ( ab, c ) ∈ ρ , for any a, b, c ∈ S . • with compatibility property if ( x, y ) , ( u, v ) ∈ ρ implies ( xu, yv ) ∈ ρ for any x, y, u, v ∈ S .In the sequel, the positive quasiorders will also be considered. Recall that the divisionrelation | on a semigroup S , defined by a | b def ⇔ ( ∃ x, y ∈ S ) b = xay, for a, b ∈ S , is the smallest positive quasi-order defined on S . The positive quasi-orderswere introduced in [46]. In [50] their link to semilattice decompositions of semigroups wasestablished. In [42] their possible applications in psychology were announced. Finally, closeconnections between positive quasiorders and subsemigroups defined above were given in[6]. Here we mention some of these results: 10 heorem 2.1. Let ρ be a quasiorder on S . The following conditions on a semigroup S are equivalent: (i) ρ is a positive quasiorder; (ii) ( ∀ a, b ∈ S ) ( ab ) ρ ⊆ aρ ∩ bρ ; (iii) ( ∀ a, b ∈ S ) ρa ∪ ρb ⊆ ρ ( ab ) ; (iv) aρ is an ideal for any a ∈ S ; (v) ρa is a convex subset of S for any a ∈ S . Theorem 2.2.
Let ρ be a quasiorder on S . The following conditions on a semigroup S are equivalent: (i) ρ is a positive quasiorder with cm-property; (ii) ρa is a convex subsemigroup of S for any a ∈ S ; (iii) ( ∀ a, b ∈ S ) ( ab ) ρ = aρ ∩ bρ . Finally, we can say that, from the point of view of the classical semigroup theory, theinterrelations between the following notions are of interest: • semilattice decomposition of semigroups, • completely isolated and convex subsemigroups and/or ideals, • positive quasiorders. Isomorphism theorems for semigroups
Let us remember that congruences on groups are uniquely determined by their normalsubgroups, and, on the other hand, there is a bijection between congruences and the idealsof rings. The study of congruences on semigroups is more complicated - no such deviceis available. One must study congruences as such. A congruence ρ on a semigroup S isan equivalence, i.e. symmetric quasiorder, with the compatibility property . Classically, thequotient set S/ρ is then provided with a semigroup structure.
Theorem 2.3.
Let S be a semigroup and ρ a congruence on it.Then S/ρ is a semigroupwith respect to the operation defined by ( xρ )( yρ ) = ( xy ) ρ , and the mapping π : S → S/ρ , π ( x ) = xρ , x ∈ S , is an onto homomorphism. Provided that the first ) isomorphism theorem for semigroups follows. Theorem 2.4.
Let f : S → T be a homomorphism between semigroups S and T . Then (i) ker f = f ◦ f − = { ( x, y ) ∈ S × S : f ( x ) = f ( y ) } is a congruence on S ; the mapping θ : S/ker f → T defined by θ ( x ( ker f )) = f ( x ) is an embedding suchthat f = θ ◦ π ; (iii) if f maps S onto T , then θ is an isomorphism. The theorem which follows is concerned with a more general situation.
Theorem 2.5.
Let ρ be a congruence on a semigroup S , and let f : S → T be a homomor-phism between semigroups S and T such that ρ ⊆ ker f . Then there exists a homomorphismof semigroups θ : S/ρ → T , such that f = θ ◦ π . If, in addition, f is onto , then θ is anisomorphism. Constructive algebra is a relatively old discipline developed among others by L. Kronecker,van der Waerden, A. Heyting. For more information on the history see [35], [51]. One of themain topics in constructive algebra is constructive algebraic structures with the relationof (tight) apartness x − exists unless we know that x is apart from zero, i.e. | x | > x = 0. Furthermore, in fields x − exists only if x is apart from 0, [3]) Thestudy of algebraic structures in the presence of tight apartness was started by Heyting,[24]. Heyting gave the theory a firm base in [26]. Roughly, the descriptive definition of astructure with apartness includes two main parts:- the notion of a certain classical algebraic structure is straightforwardly adopted;- a structure is equipped with an apartness with standard operations respecting thatapartness.Quotient structures are not part of BISH . A quotient structure does not, in general,have a natural apartness relation. So, the Quotient Structure Problem - QSP is one ofthe very first problems which has to be considered for any structure with apartness. The
QSP ’s solutions for groups with tight apartness and commutative rings with tight apartnessobtained at the beginning of the 1980s inspired us to give solutions of
QSP for sets andsemigroups with apartness in 2013, [16].A lot of ideas, notions and notations come from, for example, the constructive analysis,and, especially, from the constructive topology, as well as from constructive theories ofgroups and rings with tight apartness. Although the area of constuctive semigroups withapartness is still in its infancy, we can already conclude that, similarly to the clasical case,the semigroups with apartness do not much resemble groups and rings. In fact, they donot much resemble any other constructive algebraic structures with apartness.
It is well known that formalization is a general method in science. Although it was createdas a technique in logic and mathematics, it has entered into engineering as well. For-12al engineering methods can be understood as mathematically-based techniques for thefunctional specification, development and verification in the engineering of software andhardware systems. Despite some initial suspicion, it was proved that formal methods arepowerful enough to deal with real life systems. For example, it is shown that “softwareof the size and complexity as we find in modern cars today can be formally specified andverified by applying computer based tools for modeling and interactive theorem proving,”[14].Proof assistants are computer systems which give a user the possibility to do mathe-matics on a computer: from (numerical and symbolical) computing aspects to the aspectsof defining and proving. The latter ones, doing proofs, are the main focus. It is believedthat, besides their great future within the area of mathematics formalization, their appli-cations within computer-aided modelling and verification of the systems are and will bemore important. One of the most popular, with the intuitionistic background, is the proofassistant computer system
Coq .Coq is used for formal proves of well known mathematical theorems, such as, for exam-ple, the Fundamental Theorem of Algebra, FTA, [21]. For that purpose, the constructivealgebraic hierarchy for Coq was developed, [22], consisting of constructive basic algebraicstructures (semigroups, monoids, groups, rings, fields) with tight apartness. In addition,all these structures are limited to the commutative case. As it is noticed in [22] “thatalgebraic hierarchy has been designed to prove FTA. This means that it is not rich as onewould like. For instance, we do not have noncommutative structure because they did notoccur in our work.” ... So, a question which arises from this is:
What can be done in connection with noncommutative semigroups with apart-ness where apartness is only “ordinary” and not the tight one?
We put noncommutative constructive semigroups with “ordinary” apartness in the core ofour study, proving first, of course, that such semigroups do exist, [16]. As in [5], we made“every effort to follow classical development along the lines suggested by familiar classicaltheories or in all together new directions.”The results of our several years long investigations, [16], [17], [38], present a semigroupfacet of some relatively well established directions of constructive mathematics which, tothe best of our knowledge, have not yet been considered within the semigroup community.The initial step towards grounding the theory done through our papers will be developedthrough the scope of this paper. We are going to give a critical review of some of thoseresults as well as the solutions to some of the open problems arising from those papers.
Before starting our constructive examination of sets and semigroups with apartness, weshould clarify its setting. By constructive mathematics we mean Bishop-style mathemat-ics,
BISH . We adopt Fred Richman’s viewpoint, [44], where constructive mathematics issimply mathematics carried out with intuitionistic logic. The Bishop-style of constructive13athematics enables one to interpret the results both in classical mathematics,
CLASS ,and other varieties of constructivism. We regard classical mathematics as Bishop-stylemathematics plus the law of excluded middle,
LEM . This logical principle can be re-garded as the main source of nonconstructivity. It was Brouwer, [13], who first observedthat
LEM was extended without justification to statements about infinite sets. Severalconsequences of
LEM are not accepted in Bishop’s constructivism. We will mention twosuch nonconstructive principles - the ones which will be used latter. • The limited principle of omniscience , LPO : for each binary sequence( a n ) n ≥ , either a n = 0 for all n , or else there exists n with a n = 1. • Markov’s principle , MP : For each binary sequence ( a n ) n ≥ , if it is im-possible that a n = 0 for all n , then there exists n with a n = 1. Remark 2. LPO is equivalent to the decidability of equality on the real number line R . ∀ x ∈ R ( x = 0 ∨ x = 0) . A detailed constructive study of R can be found in [9]. Within constructive mathematics, a statement P , as in classical mathematics, can bedisproved by giving a counterexample. However, it is also possible to give a Brouweriancounterexample to show that the statement is nonconstructive. A Brouwerian counterexam-ple to a statement P is a constructive proof that P implies some nonconstructive principle,such as, for example, LEM , and its weaker versions
LPO , MP . It is not a counterexamplein the true sense of the word - it is just an indication that P does not admit a constructiveproof. More details about nonconstructive principles and various classical theorems thatare not constructively valid can be found in [28]. The cornerstones for
BISH include the notion of positive integers, sets and functions. Theset N of positive numbers is regarded as a basic set, and it is assumed that the positivenumbers have the usual algebraic and order properties, including mathematical induction.Contrary to the classical case, a set exists only when it is defined. To define a set S ,we have to give a property that enables us to construct members of S , and to describe theequality = between elements of S – which is a matter of convention, except that it mustbe an equivalence. A set ( S, =) is an inhabited set if we can construct an element of S .The distinction between the notions of a nonempty set and an inhabited set is a key inconstructive set theories. The notion of equality of different sets is not defined. The onlyway in which elements of two different sets can be regarded as equal is by requiring themto be subsets of a third set. For this reason, the operations of union and intersection aredefined only for sets which are given as subsets of a given set. There is another problemto face when we consider families of sets that are closed under a suitable operation of14omplementation. Following [5] “we do not wish to define complementation in the termsof negation; but on the other hand, this seems to be the only method available. The wayout of this awkward position is to have a very flexible notion based on the concept of a setwith apartness .”A property P , which is applicable to the elements of a set S , determines a subset of S denoted by { x ∈ S : P ( x ) } . Furthermore, we will be interested only in properties P ( x )which are extensional in the sense that for all x , x ∈ S with x = x , P ( x ) and P ( x )are equivalent. Informally, it means that “it does not depend on the particular descriptionby which x is given to us”.An inhabited subset of S × S , or, equivalently, a property applicable to elements of S × S ,is called a binary relation on S . In general, there are many properties that binary relationsmay satisfy on a given set. For instance, reflexivity, symmetry, transitivity, irreflexivity,strong irreflexivity, co-transitivity play a role under constructive rules.In CLASS , equivalence is the natural generalization of equality. A theory with equiv-alence involves equivalence and functions, and relations respecting this equivalence. Inconstructive mathematics the same works without difficulty, [45].Many sets come with a binary relation called inequality satisfying certain properties,and denoted by =, ⋍ . In general, more computational information is required todistinguish elements of a set S , than to show that elements are equal. Comparing with CLASS , the situation for inequality is more complicated. There are different types ofinequalities (denial inequality, diversity, apartness, tight apartness - to mention a few),some of them completely independent, which only in
CLASS are equal to one standardinequality. So, in
CLASS the study of the equivalence relation suffices, but in constructivemathematics, an inequality becomes a “basic notion in intuitionistic axiomatics”. Apart-ness, as a positive version of inequality, “is yet another fundamental notion developed inintuitionism which shows up in computer science,” [29].Let ( S, =) be an inhabited set. By an apartness on S we mean a binary relation S which satisfies the axioms of strong irreflexivity, symmetry and cotransitivity:(Ap1) ¬ ( x x )(Ap2) x y ⇒ y x ,(Ap3) x z ⇒ ∀ y ( x y ∨ y z ) . If x y , then x and y are different, or distinct. Roughly speaking, x = y means that wehave a proof that x equals y while x y means that we have a proof that x and y aredifferent. Therefore, the negation of x = y does not necessarily imply that x y and viceversa: given x and y , we may have neither a proof that x = y nor a proof that x y .The negation of apartness is an equivalence ( ≈ ) = def ( ¬ weak equality on S . Remark 3.
The statement that every equivalence relation is the negation of some apart-ness relation is equivalent to the excluded middle. The statement that the negation of anequivalence relation is always an apartness relation is equivalent to the nonconstructive deMorgan law. S is tight if(Ap4) ¬ ( x y ) ⇒ x = y .Apartness is tight just when ≈ and = are the same, that is ¬ ( x y ) ⇔ x = y .In some books and papers, such as [51], the term “preapartness” is used for an apartnessrelation, while “apartness” means tight apartness. The tight apartness on the real numberswas introduced by L. E. J. Brouwer in the early 1920s. Brouwer introduced the notionof apartness as a positive intuitionistic basic concept. A formal treatment of apartnessrelations began with A. Heyting’s formalization of elementary intuitionistic geometry in[25]. The intuitionistic axiomatization of apartness is given in [27].By extensionality, we have(Ap5) x y ∧ y = z ⇒ x z ,the equivalent form of which is(Ap5’) x y ∧ x = x ′ ∧ y = y ′ ⇒ x ′ y ′ .A set with apartness ( S, = , S . The existence of an apartness relation on a structure often gives riseto an apartness relation on another structure. For example, given two sets with apartness( S, = S , S ) and ( T, = T , T ), it is permissible to construct the set of mappings betweenthem. Let f : S → T be a mapping (function) of sets with apartness. The well-definedness or weak extensionality of f , i.e. ∀ x,y ∈ S ( x = S y ⇒ f ( x ) = T f ( y )) , follows by extensionality. A mapping f : S → T is:- onto S or surjection : ∀ y ∈ T ∃ x ∈ S ( y = T f ( x ));- one-one or injection : ∀ x,y ∈ S ( f ( x ) = T f ( y ) ⇒ x = S y );- bijection between S and T : it is a one-one and onto.Constructively, as apartness is more fundamental than equality, the property of strongextensionality is more fundamental than the well-definedness. A mapping f : S → T is an strongly extensional mapping, or, for short, an se-mapping , if ∀ x,y ∈ S ( f ( x ) T f ( y ) ⇒ x S y ) . Furthermore, f is- apartness injective , shortly a-injective : ∀ x,y ∈ S ( x S y ⇒ f ( x ) T f ( y ));- apartness bijective : a-injective, se-bijective.Given two sets with apartness S and T it is permissible to construct the set of orderedpairs ( S × T, = , s, t ) u, v ) def ⇔ s S u ∨ t T v. .1.1 Distinguishing subsets The presence of apartness implies the appearance of different types of substructures con-nected to it. Inspired by the constructive topology with apartness [12], we define therelation ⊲⊳ between an element x ∈ S and a subset Y of S by x ⊲⊳ Y def ⇔ ∀ y ∈ Y ( x y ) . A subset Y of S has two natural complementary subsets: the logical complement of Y ¬ Y def = { x ∈ S : x / ∈ Y } , and the apartness complement or, shortly, the a-complement of Y ∼ Y def = { x ∈ S : x ⊲⊳ Y } . Denote by e x the a-complement of the singleton { x } . Then it can be easily shown that x ∈∼ Y if and only if Y ⊆ e x. If the apartness is not tight we can find subsets Y with ∼ Y ⊂ ¬ Y as in the followingexample. Example 1.
Let S = { a, b, c } be a set with apartness defined by { ( a, c ) , ( c, a ) , ( b, c ) , ( c, b ) } and let Y = { a } . Then the a-complement ∼ Y = { c } is a proper subset of its logicalcomplement ¬ Y = { b, c } . For a tight apartness, the two complements are constructive counterparts of the classicalcomplement. In general, we have ∼ Y ⊆ ¬ Y .
However, even for a tight apartness, theconverse inclusion entails the Markov principle. This result illustrates a main featureof constructive mathematics: classically equivalent notions could be no longer equivalentconstructively. For which type of subset of a set with apartness do we have equality betweenits two complements? It turns out that the answer initiated a development of order theory for sets and semigroups with apartnessThe complements are used for the classification of subsets of a given set. A subset Y of S is • a detachable subset in S or, in short, a d-subset in S if ∀ x ∈ S ( x ∈ Y ∨ x ∈ ¬ Y ); • a strongly detachable subset of S , shortly an sd-subset of S , if ∀ x ∈ S ( x ∈ Y ∨ x ∈ ∼ Y ) , • a quasi-detachable subset of S , shortly a qd-subset of S , if ∀ x ∈ S ∀ y ∈ Y ( x ∈ Y ∨ x y ) . Theorem 3.1.
Let Y be a subset of S . Then: (i) Any sd-subset is a qd-subset of S . The converse implication entails LPO . (ii) Any qd-subset Y of S satisfies ∼ Y = ¬ Y . (iii) If any qd-subset is a d-subset, then
LPO holds. (iv)
If any d-subset is a qd-subset, then MP holds. (v) Any sd-subset is a d-subset of S . The converse implication entails MP . (vi) If any subset of a set with apartness S is a qd-subset, then LPO holds.Proof. (i). Let Y be an sd-subset of S . Then, applying the definition and logical axiomwe have ∀ x ∈ S ( x ∈ Y ∨ x ∈ ∼ Y ) ⇔ ∀ x ∈ S ( x ∈ Y ∨ ∀ y ∈ Y ( x y )) ⇒ ∀ x ∈ S ∀ y ∈ Y ( x ∈ Y ∨ x y ) . In order to prove the second part of this statement, we consider the real number set R with the usual (tight) apartness and the subset Y = e . Then, for each real number x and for each y ∈ Y it follows, from the co-transitivity of y x or x x ∈ Y or x y . Consequently, Y is a qd-subset of R . On the other hand, if Y is ansd-subset of R , then for each x ∈ R , either x ∈ Y or x ∈∼ Y. In the former case, x x = 0, hence LPO holds.(ii). Let Y be a qd-subset, and let a ∈ ¬ Y . By assumption we have ∀ x ∈ S ∀ y ∈ Y ( x ∈ Y ∨ x y ) , so substituting a for x , we get ∀ y ∈ Y ( a ∈ Y ∨ a y ), and since, by assumption, ¬ ( a ∈ Y ),it follows that a y for all y ∈ Y . Hence a ∈∼ Y .(iii). Let S be the real number set R with the usual apartness e R . If e R , then x ∈ e ¬ ( x ∈ e x . In the latter case ¬ ( x x = 0 . Thus we obtainthe property ∀ x ∈ R ( x ∨ x = 0) which, in turn, is equivalent to LPO .(iv). Consider a real number a with ¬ ( a = 0) and let S be the set { , a } endowed withthe usual apartness of R . For Y = { } , since 0 ∈ Y and a ∈ ¬ Y, it follows that Y is ad-subset of S . On the other hand, if Y is a qd-subset of S , then a ¬ ( a = 0), a MP .18v). The first part follows immediately from (i), (ii) and the definition of d-subsets.The converse follows from (i) and (iv).(vi). Consider again R with the usual apartness and define Y = { } . If Y is a qd-subsetof R , then for all x ∈ R we have x = 0 or x LPO holds.If the apartness is not tight, we can find subsets which are not qd-subsets, let alonesd-subsets. To show this, let us consider the set S = { a, b, c } with the apartness defined inExample 1 and define Y = { a } . Then Y is not a qd-subset of S . If we work with a tightapartness, although vacuously true in classical mathematics, the properties of detachabilityare not automatically satisfied in BISH . The Brouwerian examples from Theorem 3.1motivate the use of qd-subsets. Constructive mathematics brings to the light some notionswhich are invisible to the classical eye (here, the three notions of detachability).
Let ( S × S, = , S × S , or, equivalently,a property applicable to the elements of S × S , is called a binary relation on S . Let α bea relation on S . Then ( a, b ) ⊲⊳ α ⇔ ∀ ( x,y ) ∈ α (( a, b ) x, y )) , for any ( a, b ) ∈ S × S . The apartness complement of α is the relation ∼ α = { ( x, y ) ∈ S × S : ( x, y ) ⊲⊳ α } . In general, we have ∼ α ⊆ ¬ α , which is shown by the following example. Example 2.
Let S = { a, b, c } be a set with apartness defined by { ( a, c ) , ( c, a ) , ( b, c ) , ( c, b ) } .Let α = { ( a, c ) , ( c, a ) } be a relation on S . Its a-complement ∼ α = { ( a, a ) , ( b.b ) , ( c, c ) , ( a, b ) , ( b, a ) } is a proper subset of its logical complement ¬ α . The relation α defined on a set with apartness S is • irreflexive if ∀ x ∈ S ¬ (( x, x ) ∈ α ); • strongly irreflexive if ( x, y ) ∈ α ⇒ x y ; • co-transitive if ( x, y ) ∈ α ⇒ ∀ z ∈ S (( x, z ) ∈ α ∨ ( z, y ) ∈ α ) . It is easy to check that a strongly irreflexive relation is also irreflexive. For a tightapartness, the two notions of irreflexivity are classically equivalent but not so constructively.More precisely, if each irreflexive relation were strongly irreflexive then MP would hold.In the constructive order theory, the notion of co-transitivity, that is the propertythat for every pair of related elements, any other element is related to one of the originalelements in the same order as the original pair is a constructive counterpart to classicaltransitivity, [16]. 19 emma 3.1.
Let α be a relation on S . Then: (i) α is strongly irreflexive if and only if ∼ α is reflexive; (ii) if α is reflexive then ∼ α is strongly irreflexive; (iii) if α is symmetric then ∼ α is symmetric; (iv) α is co-transitive then ∼ α is transitive.Proof. (i). Let α be a strongly irreflexive relation on S . For each a ∈ S , it can be easilyproved that ( a, a ) x, y ) for all ( x, y ) ∈ α .Let ∼ α be reflexive, that is ( x, x ) ∈∼ α , for any x ∈ S . On the other hand, thedefinition of the a-complement implies ( x, y ) x, x ) for any ( x, y ) ∈ α . So, x x or x y .Thus, x y , that is, α is strongly irreflexive.(ii). Let α be reflexive. Let ( x, y ) be an element of ∼ α . Since α is reflexive, ( y, y ) ∈ α hence ( x, y ) y, y ) which implies x y. Consequently, ∼ α is strongly irreflexive.(iii). If α is symmetric, then( x, y ) ∈∼ α ⇔ ∀ ( a,b ) ∈ α (( x, y ) a, b )) ⇒ ∀ ( b,a ) ∈ α (( x, y ) b, a )) ⇒ ∀ ( b,a ) ∈ α ( x b ∨ y a ) ⇒ ∀ ( a,b ) ∈ α (( y, x ) a, b )) ⇔ ( y, x ) ∈∼ α. (iv). If ( x, y ) ∈∼ α and ( y, z ) ∈∼ α , then, by the definition of ∼ α , we have that( x, y ) ⊲⊳ α and ( y, z ) ⊲⊳ α . For an element ( a, b ) ∈ α , by co-transitivity of α , we have( a, x ) ∈ α or ( x, y ) ∈ α or ( y, z ) ∈ α or ( z, b ) ∈ α . Thus ( a, x ) ∈ α or ( z, b ) ∈ α , whichimplies that a x or b z , that is ( x, z ) a, b ). So, ( x, z ) ⊲⊳ α and ( x, z ) ∈ ∼ α . Therefore, ∼ α is transitive. Remark 4.
As it is shown in Lemma 3.1, it can be proved that the logical complement ofeach co-transitive relation is transitive. However, if the logical complement of any transitiverelation were co-transitive, then MP would hold. The apartness complement ∼ α of a relation α of S can be transitive without assumingco-transitivity of α . So, the converse statement from Lemma 3.1(iv), in general, is nottrue. Example 3.
Let ( S, = , be a set with apartness defined in Example 2.(1.) A strongly irreflexive (symmetric) relation α = { ( a, c ) , ( c, a ) } , which is not co-transitive has the a-complement ∼ α = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } which is transitive. α = { ( a, c ) , ( c, a ) , ( b, c ) } , which isnot co-transitive, has the a-complement ∼ α = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } which is transitive. A relation τ defined on a set with apartness S is a • weak co-quasiorder if it is irreflexive and cotransitive, • co-quasiorder if it is strongly irreflexive and cotransitive. Remark 5. “One might expect that the splitting of notions leads to an enormous pro-liferation of results in the various parts of constructive mathematics when compared withtheir classical counterparts. In particular, usually only very few constructive versions of aclassical notion are worth developing since other variants do not lead to a mathematicallysatisfactory theory,” [51].Even if the two classically (but not constructively) equivalent variants of a co-quasiorderare constructive counterparts of a quasiorder in the case of (a tight) apartness, the strongervariant, co-quasiorder, is, of course, the most appropriate for a constructive development ofthe theory of semigroups with apartness we develop, which will be evident in the continuationof this paper. The weaker variant, that is, weak co-quasiorder, could be relevant in analysis.
As in Example 2 the a-complement of a relation can be a proper subset of its logicalcomplement. If the relation in question is a co-quasiorder, then we have the followingimportant properties.
Proposition 3.1.
Let τ be a co-quasiorder on S . Then: (i) τ is a qd-subset of S × S ; (ii) ∼ τ = ¬ τ .Proof. (i). Let ( x, y ) ∈ S × S . Then, for all ( a, b ) ∈ τ , aτ x ∨ xτ b ⇒ aτ x ∨ xτ y ∨ yτ b ⇒ a x ∨ xτ y ∨ y b ⇒ ( a, b ) x, y ) ∨ xτ y, that is, τ is a qd-subset.(ii). It follows from (i) and Proposition 3.1(ii).The co-quasiorder is one of the main building blocks for the order theory of semigroupswith apartness we develop.In general, to describe the relation we have to determine which ordered pairs belong to τ , that is, we have to determine aτ and τ a , the left and the right τ -class of each element a from S . That is the way to connect (in CLASS and in
BISH as well) a relation definedon a given set with certain subsets of the set. Starting from an sd-subset T of S , we areable to construct co-quasiorders as follows. 21 emma 3.2. Let T be an sd-subset of a set with apartness S . Then, the relation τ on S ,defined by ( a, b ) ∈ τ def ⇔ a ∈∼ T ∧ b ∈ T, is a co-quasiorder on S .Proof. Let ( a, b ) ∈ τ , that is a ∈∼ T and b ∈ T , and let x ∈ S . By the assumption, T isan sd-subset, so we have x ∈ T or x ∈∼ T . If x ∈ T , then, by the definition of τ , we have( a, x ) ∈ τ . Similarly, if x ∈∼ T , then ( x, b ) ∈ τ . Thus, co-transitivity of τ is proved. Bythe definition of τ , the strong irreflexivity follows immediately. Thus, τ is a co-quasiorderon S . Example 4.
Let S = { a, b, c, d, e } be a set with the diagonal △ S = { ( a, a ) , ( b, b ) , ( c, c ) , ( d, d ) , ( e, e ) } as the equality relation. If we denote by K the set △ S ∪ { ( a, b ) , ( b, a ) } , then we can definean apartness on S to be ( S × S ) \ K . Thus, ( S, = , is a set with apartness. Therelation τ ⊆ S × S , defined by τ = { ( c, a ) , ( c, b ) , ( d, a ) , ( d, b ) , ( d, c ) , ( e, a ) , ( e, b ) , ( e, c ) , ( e, d ) } , is a co-quasiorder on S . (Left) τ -classes of S are: aτ = bτ = ∅ , cτ = { a, b } , dτ = { a, b, c } , eτ = { a, b, c, d } . It can be easily checked that all those τ -classes are sd-subsets of S . Generally speaking, for a co-quasiorder defined on a set with apartness we can notprove that its left and/or right classes are d-subsets or sd-subsets. More precisely, we canprove the following result.
Proposition 3.2.
Let τ be a co-quasiorder. Then: (i) if aτ is a d-subset of S for any a ∈ S , then LPO holds; (ii) if aτ is an sd-subset of S for any a ∈ S , then LPO holds.
Proof. (i). Similar to the proof of Theorem 3.1(iii). It suffices to let τ be the usualapartness on the real number set and a = 0 . (ii). We can use the same example as above and apply Theorem 3.1(i).Having in mind what is just proved, we cannot expect to prove Lemma 3.2 from [17], asstated in [17], with d-subsets or sd-subsets without the Constant Domain Axiom, CDA . Intuitionistic logic of constant domains CD as a background
Following [1], the intuitionistic logic of constant domains CD arises from a very naturalKripke-style semantics, which was proposed in [20] as a philosophically plausible interpreta-tion of intuitionistic logic. CD can be formalized as intuitionistic logic extended with from22he classical algebra point of view pretty strong principle, the Constant Domain Axiom,CDA, (cid:13) ∀ x ( P ∨ R ( x )) → ( P ∨ ∀ x R ( x )) , where x is not a free variable of P . The intermediate logic obtained in this way, as it ispointed out in [1], further proves intuitionistically as well as classically valid theorems, yetthey often possess a strong constructive flavour.From a given co-quasiorder τ , with CD as a logical background, we are able to provethe connection of its classes with sd-subsets of S . Lemma 3.3.
Let τ be a co-quasiorder on a set S . Then aτ (respectively) τ a ) is an sd-subset of S , such that a ⊲⊳ aτ (respectively a ⊲⊳ τ a ), for any a ∈ S . Moreover, if ( a, b ) ∈ τ ,then aτ ∪ τ b = S is true for all a, b ∈ S .Proof. Let x ∈ S and y ∈ aτ . Then, by the co-transitivity of τ , we have ( a, x ) ∈ τ or( x, y ) ∈ τ . So, x ∈ aτ or, by strong irreflexivity, x y . Thus, aτ is an sd-subset of S , and,again by strong irreflexivity, we have a ⊲⊳ aτ . In a similar manner, we can prove that τ a is an sd-subset of S such that a ⊲⊳ τ a .Let a, b ∈ S such that ( a, b ) ∈ τ , and let x ∈ S . This means, by co-transitivity, that( a, x ) ∈ τ or ( x, b ) ∈ τ , i.e. x ∈ τ a or x ∈ τ b , or, equivalently, x ∈ aτ ∪ τ b . Therefore, S ⊆ aτ ∪ τ b , which implies the equality. Remark 6.
In intuitionistic logic of constant domain CD , the notions of sd-subset andqd-subset coincide. The Quotient Structure Problem,
QSP , is one of the very first problems which has to beconsidered for any structure with apartness. The solutions of the
QSP problem for sets andsemigroups with apartness was for the first time given in [16]. Those results are improvedin [38]. In what follows, we achieve a little progress in that direction. Theorem 3.2,the key theorem for the
QSP ’s solution generalizes the similar ones from [16], [38]. Inaddition, as a generalization of the Theorem 3.3, the first apartness isomorphism theorem,the new Theorem 3.4, that we call the second apartness isomorphism theorem for sets withapartness is formulated and proved.The quotient structures are not part of
BISH . A quotient structure does not have, ingeneral, a natural apartness relation. For most purposes, we overcome this problem using a co-equivalence –symmetric co-quasiorder–instead of an equivalence. Existing propertiesof a co-equivalence guarantee its a-complement is an equivalence as well as the quotient setof that equivalence will inherit an apartness. The following notion will be necessary. Forany two relations α and β on S we can define a relation Π on S/β by ( xβ ) Π ( yβ ) if andonly if ( x, y ) ∈ α . We, further, say that α defines an apartness on S /β if Π is an apartnesson S/β , that is if we have(Ap6) xβ yβ def ⇔ ( x, y ) ∈ α x, a ) ∈ β ∧ ( y, b ) ∈ β ) ⇒ (( x, y ) ∈ α ⇔ ( a, b ) ∈ α ) . The next theorem is the key for the solution of
QSP for sets with apartness. Itgeneralizes the results from [16], [38].
Theorem 3.2.
Let S be a set with apartness. Then: (i) Let ε be an equivalence, and κ a co-equivalence on S . Then, κ defines an apartnesson the factor set S/ε if and only if ε ∩ κ = ∅ . (ii) The quotient mapping π : S → S/ε , defined by π ( x ) = xε , is an onto se-mapping.Proof. (i). Let x, y ∈ S and assume that ( x, y ) ∈ ε ∩ κ . Then ( x, y ) ∈ ε and ( y, y ) ∈ ε ,which, by extensionality (Ap6’) of κ , and ( x, y ) ∈ κ gives ( y, y ) ∈ κ , which is impossible.Thus, ε ∩ κ = ∅ .Let ( x, a ) , ( y, b ) ∈ ε and ( x, y ) ∈ κ . Then, by co-transitivity of κ and by assumption,we have ( x, y ) ∈ κ ⇒ ( x, a ) ∈ κ ∨ ( a, y ) ∈ κ ⇒ ( x, a ) ∈ κ ∨ ( a, b ) ∈ κ ∨ ( b, y ) ∈ κ ⇒ ( a, b ) ∈ κ. (ii). Let π ( x ) π ( y ), that is xε yε , which, by (i), means that ( x, y ) ∈ κ . Then, by thestrong irreflexivity of κ , we have x y . So π is an se-mapping.Let aε ∈ S/ ∼ κ and x ∈ aε . Then ( a, x ) ∈∼ κε , i.e. aε = xε , which implies that aε = xε = π ( x ). Thus π is an onto mapping. Corollary 3.1. If κ is a co-equivalence on S , then the relation ∼ κ (= ¬ κ ) is an equivalenceon S , and κ defines an apartness on S/ ∼ κ .Proof. By Lemma 3.1, ∼ κ is an equivalence, by Proposition 3.1, ( ∼ κ ) = ( ¬ κ ), and, byTheorem 3.2, κ defines an apartness on S/ ∼ κ .Let f : S → T be an se-mapping between sets with apartness. Then the relationcoker f def = { ( x, y ) ∈ S × S : f ( x ) f ( y ) } defined on S is called the co-kernel of f . Now, the first apartness isomorphism theorem for sets with apartness follows. Theorem 3.3.
Let f : S → T be an se-mapping between sets with apartness. Then (i) the co-kernel of f is a co-equivalence on S which defines an apartness on S/ ker f ; (ii) the mapping θ : S/ ker f → T , defined by θ ( x (ker f )) = f ( x ) , is a one-one, a-injective se-mapping such that f = θ ◦ π ; if f maps S onto T , then θ is an apartness bijection.Proof. See [38].Now, the second apartness isomorphism theorem , a generalised version of Theorem 3.3,for sets with apartness follows.
Theorem 3.4.
Let f : S → T be a mapping between sets with apartness, and let κ be aco-equivalence on S such that κ ∩ ker f = ∅ . Then: (i) κ defines apartness on factor set S/ ker f ; (ii) the projection π : S → S/ ker f defined by π ( x ) = x (ker f ) is an onto se-mapping; (iii) the mapping f induces a one-one mapping θ : S/ ker f → T given by θ ( x (ker f )) = f ( x ) , and f = θ ◦ π ; (iv) θ is an se-mapping if and only if coker f ⊆ κ ; (v) θ is a-injective if and only if κ ⊆ coker f .Proof. (i). It follows from Theorem 3.2(i).(ii). It follows from Theorem 3.2(ii).(iii). This was shown in Theorem 3.3.(iv). Let θ be an se-mapping. Let ( x, y ) ∈ coker f for some x, y ∈ S . Then, bydefinition of coker f and θ , the assumption and (i), we have f ( x ) f ( y ) ⇔ θ ( x (ker f )) θ ( y (ker f )) ⇒ x (ker f ) y (ker f ) ⇔ ( x, y ) ∈ κ. Conversely, let coker f ⊆ κ . By assumption, (i), and the definitions of θ and coker f ,we have θ ( x (ker f )) θ ( y (ker f )) ⇔ f ( x ) f ( y ) ⇔ ( x, y ) ∈ coker f ⇒ ( x, y ) ∈ κ ⇔ x (ker f ) y (ker f ) . (v). Let θ be a-injective, and let ( x, y ) ∈ κ . Then, by (i), we have x (ker f ) y (ker f ) ⇒ θ ( x (ker f )) θ ( y (ker f )) ⇔ f ( x ) f ( y ) ⇔ ( x, y ) ∈ coker f. κ ⊆ coker f . Then x (ker f ) y (ker f ) ⇔ ( x, y ) ∈ κ ⇒ ( x, y ) ∈ coker f ⇔ f ( x ) f ( y ) ⇔ θ ( x (ker f )) θ ( y (ker f )) . Corollary 3.2.
Let f : S → T be a mapping between sets with apartness, and let κ be aco-equivalence on S such that κ ∩ ker f = ∅ . Then: (i) f is an se-mapping if and only if ker f is strongly irreflexive; (ii) if θ : S/ ker f → T , defined by θ ( x (ker f )) = f ( x ) , is an se-mapping, then f is anse-mapping too.Proof. (i). Let f be an se-mapping. Then, by Theorem 3.3, ker f is strongly irreflexive.The converse is almost obvious.(ii). If θ is an se-mapping then, by Theorem 3.4(iv), we have that coker f ⊆ κ . So, thestrong irreflexivity of κ implies the strong irreflexivity of coker f , which, by (i), implies f is an se-mapping. Given a set with apartness ( S, = , S, = , , · ) is a semigroup with apartness if the binary operation · is associative(A) ∀ a,b,c ∈ S [( a · b ) · c = a · ( b · c )],and strongly extensional(S) ∀ a,b,x,y ∈ S ( a · x b · y ⇒ ( a b ∨ x y )).As usual, we are going to write ab instead of a · b . For example, for a given set withapartness A we can construct a semigroup with apartness S = A A in the following way. Theorem 3.5.
Let S be the set of all se-functions from A to A with the standard equality = f = g ⇔ ∀ x ∈ A ( f ( x ) = g ( x )) and apartness f g ⇔ ∃ x ∈ A ( f ( x ) g ( x )) . Then ( S, = , , ◦ ) is a semigroup with respect to the binary operation ◦ of composition offunctions. roof. See [17].Until the end of of this paper, we adopt the convention that semigroup means semigroupwith apartness . Apartness from Theorem 3.5 does not have to be tight, [16].Let S and T be semigroups with apartness. A mapping f : S → T is a homomorphismif ∀ x,y ∈ S ( f ( xy ) = f ( x ) f ( y )) . A homomorphism f is- an se-embedding if it is one-one and strongly extensional;- an apartness embedding if it is a-injective se-embedding;- an apartness isomorphism if it is apartness bijection and se-homomorphism.Within CLASS , the semigroups can be viewed, historically, as an algebraic abstractionof the properties of the composition of transformations on a set. Cayley’s theorem forsemigroups (which can be seen as an extension of the celebrated Cayley’s theorem ongroups) stated that every semigroup can be embedded in a semigroup of all self-maps ona set. As a consequence of the Theorem 3.5, we can formulate the constructive Cayley’stheorem for semigroups with apartness as follows.
Theorem 3.6.
Every semigroup with apartness se-embeds into the semigroup of all stronglyextensional self-maps on a set.Proof.
See [17].
Remark 7.
Following [44], the term “constructive theorem” refers to a theorem with con-structive proof. A classical theorem that is proven in a constructive manner is a constructivetheorem.
It is a pretty common point of view that classical theorem becomes more enlighteningwhen it is seen from the constructive viewpoint. On the other hand, it can not be said thatthe theory of constructive semigroups with apartness aims at revising the whole classicalframework in nature.
We are going to encounter sd-subsets or sd-subsemigroups which have some of the prop-erties mentioned in Section 2.1.1. A strongly detachable convex (respectively completelyisolated) subsemigroup of S is called, in short, an sd-convex (respectively sd-completelyisolated) subsemigroup of S . Similarly, there are sd-convex and sd-completely isolatedideals of S . Lemma 3.4.
Let S be a semigroup with apartness. The following conditions are true: (i) Let T be an sd-convex subset of a semigroup with apartness S . If ∼ T is inhabited,then it is an ideal of S . If I is an sd-completely isolated ideal of a semigroup with apartness S , then ∼ I is aconvex subsemigroup of S .Proof. (i). Let x, y, ∈ ∼ T . Let a ∈ ∼ T and x ∈ S . By the assumption we have that ax ∈ T or ax ∈∼ T . If ax ∈ T , then, as T is convex, we have a ∈ T , which is impossible.Similarly, one can prove that xa ∈ ∼ T . So, ∼ T is an ideal of S .(ii). In a similar manner as in (i) we can prove that ∼ I is a subsemigroup of S .Let xy ∈ ∼ I . By the assumption, we have x ∈ I or x ∈∼ I . If x ∈ I , then, as I isan ideal, we have xy ∈ I , which is impossible. Thus x ∈∼ I . Similarly, we can prove that y ∈ ∼ I . So, ∼ I is convex.Let us start with an example of a co-quasiorder defined on a semigroup with apartness S . Example 5.
Let S be a semigroup given by · a b c d ea b b d d db b b d d dc d d c d cd d d d d de d d c d cLet the equality on S be the diagonal △ S = { ( a, a ) , ( b, b ) , ( c, c ) , ( d, d ) , ( e, e ) } . If we denoteby K = △ S ∪ { ( a, b ) , ( b, a ) } , then we can define an apartness on S by ( S × S ) \ K . Therelation τ ⊆ S × S , defined by τ = { ( c, a ) , ( c, b ) , ( d, a ) , ( d, b ) , ( d, c ) , ( e, a ) , ( e, b ) , ( e, c ) , ( e, d ) } , is a co-quasiorder on S . Let τ be a co-quasiorder defined on a semigroup S with apartness. Following theclassical results as much as possible, we can start with the following definition.A co-quasiorder τ on a semigroup S is • complement positive if ( a, ab ) , ( a, ba ) ∈∼ τ for any a, b ∈ S , • with constructive common multiple property , or, in short, with constructive cm-property if ( ab, c ) ∈ τ ⇒ ( a, c ) ∈ τ ∨ ( b, c ) ∈ τ for all a, b, c ∈ S , • with complement common multiple property , or, in short, with complement cm-property if ( a, c ) , ( b, c ) ∈∼ τ ⇒ ( ab, c ) ∈∼ τ for all a, b, c ∈ S .Recall, by the Proposition 3.1, ( ∼ τ ) = ( ¬ τ ). Example 6.
The co-quasiorder α defined on the semigroup S considered in Example 5 isnot complement positive because we have ( e, ea ) = ( e, d ) ∈ α . xample 7. Let S be the three element semilattice given by · a b ca a c cb c b cc c c cLet the equality on S be the diagonal △ S = { ( a, a ) , ( b, b ) , ( c, c ) } . We can define an apartness on S to be ( S × S ) \ △ S . Thus, ( S, = , , · ) is a semigroup with apartness. The relation τ ⊆ S × S , defined by τ = { ( a, b ) , ( c, a ) , ( c, b ) } , is a complement positive co-quasiorder on S .On the other hand, from ( ab, a ) = ( c, a ) ∈ τ neither ( a, a ) nor ( b, a ) are in τ , so τ does not have the constructive cm-property. From ( a, a ) ⊲⊳ τ and ( b, a ) ⊲⊳ τ , we have ( ab, a ) = ( c, a ) ∈ τ , and τ does not have the complement cm-property as well. The following lemma shows how some sd-subsets lead us to positive co-quasiorders.
Lemma 3.5.
Let S be a semigroup with apartness S . (i) If K is an sd-convex subset of S , then the relation τ defined by ( a, b ) ∈ τ def ⇔ a ∈∼ K ∧ b ∈ K is a complement positive co-quasiorder on S . (ii) If J is an sd-ideal of S such that J ⊂ S , then the relation τ defined by ( a, b ) ∈ τ def ⇔ a ∈ J ∧ b ∈∼ J is a complement positive co-quasiorder on S .Proof. (i). By Lemma 3.2, τ is a co-quasiorder on S . Let ( x, y ) ∈ τ . By the co-transitivityof τ , we have ( x, a ) ∈ τ ∨ ( a, ab ) ∈ τ ∨ ( ab, y ) ∈ τ , for any a, b ∈ S . If ( a, ab ) ∈ τ , then,by the definition of τ , we have a ∈∼ K and ab ∈ K , and, as K is a convex subset, wehave a ∈ K and b ∈ K , which is impossible. So, we have ( x, a ) ∈ τ ∨ ( ab, y ) ∈ τ . By thestrong irreflexivity of τ we have x a ∨ ab y , i.e. ( x, y ) a, ab ). Thus, we have provedthat ( a, ab ) ⊲⊳ τ for any a, b ∈ S . The proof of ( a, ba ) ⊲⊳ τ is similar. Therefore, τ is acomplement positive co-quasiorder on S .(ii). By Lemma 3.2, τ is a co-quasiorder on S . Let ( x, y ) ∈ τ . By the co-transitivityof τ , we have ( x, a ) ∈ τ ∨ ( a, ab ) ∈ τ ∨ ( ab, y ) ∈ τ, for any a, b ∈ S . If ( a, ab ) ∈ τ , then,by the definition of τ , we have a ∈ J and ab ∈∼ J , which, as J is an ideal, further implies ab ∈ J , which is a contradiction. The rest of the proof is similar to the arguments in theproof of (i). 29y Proposition 3.2, if any left/right-class of a co-quasiorder defined on a set withapartness is a (strongly) detachable subset, then LPO holds. This shows that Theorem4.1 on a complement positive co-quasiorder (and Lemma 3.2 important for its proof) from[17] cannot be proved outside intuitionistic logic of constant domains CD . Nevertheless,we can prove within intuitionistic logic the next theorem, which is its weaker version,and another important result of this section. The description of a complement positiveco-quasiorder via its classes follows. Theorem 3.7.
Let τ be a co-quasiorder τ on a semigroup S . (i) If τ is complement positive, then ∀ a,b ∈ S ( τ ( ab ) ⊆ τ a ∩ τ b ) . (ii) If τ a is an sd-ideal of S and a ⊲⊳ τ a for every a ∈ S , then τ is complement positiveand ∀ a,b ∈ S ( aτ ∪ bτ ⊆ ( ab ) τ ) . (iii) If aτ is an sd-convex subset of S , and a ⊲⊳ aτ for every a ∈ S , then τ is a complementpositive co-quasiorder.Proof. (i). Let τ be a complement positive co-quasiorder. For all a, b, x ∈ S such that x ∈ τ ( ab ), that is ( x, ab ) ∈ τ , by the co-transitivity of τ , we have(( x, a ) ∈ τ ∨ ( a, ab ) ∈ τ ) ∧ (( x, b ) ∈ τ ∨ ( b, ab ) ∈ τ ) . But τ is positive, so that we have ( x, a ) ∈ τ ∧ ( x, b ) ∈ τ , i.e. x ∈ τ a ∩ τ b .(ii). Let ( x, y ) ∈ τ and a, b ∈ S . Then, by the co-transitivity of τ ,( x, a ) ∈ τ ∨ ( a, ab ) ∈ τ ∨ ( ab, y ) ∈ τ. If a ∈ τ ( ab ), then, as aτ is an ideal, we have ab ∈ τ ( ab ), which is, by the assumption, impos-sible. Now, by the strong irreflexivity of τ , we have x a or ab y , that is ( x, y ) a, ab ).Thus, ( a, ab ) ⊲⊳ τ for any a, b ∈ S . ( a, ba ) ⊲⊳ τ can be proved similarly. Thus, τ is acomplement positive co-quasiorder.Let x ∈ aτ ∪ b , x ∈ S . By the co-transitivity and complement positivity of τ , we have x ∈ aτ ∪ bτ ⇔ x ∈ aτ ∨ x ∈ bτ ⇔ ( a, x ) ∈ τ ∨ ( b, x ) ∈ τ ⇒ (( a, ab ) ∈ τ ∨ ( ab, x ) ∈ τ ) ∨ (( b, ab ) ∈ τ ∨ ( ab, x ) ∈ τ ) ⇒ ( ab, x ) ∈ τ ⇔ x ∈ ( ab ) τ. (iii). Let ( x, y ) ∈ τ . Then, by the co-transitivity of τ ,( x, a ) ∈ τ ∨ ( a, ab ) ∈ τ ∨ ( ab, y ) ∈ τ, a, b ∈ S . Let ( a, ab ) ∈ τ , that is ab ∈ aτ . Then, by assumption, a ∈ aτ (and b ∈ aτ ), which is impossible. Now, by the strong irreflexivity of τ , we have x a or ab y ,that is, ( x, y ) a, ab ). Thus ( a, ab ) ⊲⊳ τ for any a, b ∈ S . ( a, ba ) ⊲⊳ τ can be provedsimilarly. Thus, τ is a complement positive co-quasiorder. Theorem 3.8.
A complement positive co-quasiorder with the constructive cm-property hasthe complement cm-property.Proof.
Let τ be a complement positive co-quasiorder with the constructive cm-property ona semigroup S and let a, b, c, x, y ∈ S be such that ( a, c ) , ( b, c ) ⊲⊳ τ and ( x, y ) ∈ τ . Thenwe have( x, y ) ∈ τ ⇒ ( x, ab ) ∈ τ ∨ ( ab, c ) ∈ τ ∨ ( c, y ) ∈ τ by co-transitivity ⇒ x ab ∨ ( a, c ) ∈ τ ∨ ( b, c ) ∈ τ ∨ c y by strong reflexivityand by constructive cm-property ⇒ ( ab, c ) x, y ) since ( a, c ) ⊲⊳ τ and ( b, c ) ⊲⊳ τ .Hence ( ab, c ) ⊲⊳ τ , i.e. ( ab, c ) ∈∼ τ . Intuitionistic logic of constant domains CD as a background
If we have a complement positive co-quasiorder τ on a semigroup with apartness S , wecan construct special subsets and semigroups mentioned above. Some other criteria for aco-quasiorder to be complement positive will be given too. Theorem 3.9.
The following conditions for a co-quasiorder τ on a semigroup S are equiv-alent: (i) τ is complement positive; (ii) ∀ a,b ∈ S ( aτ ∪ bτ ⊆ ( ab ) τ ) ; (iii) ∀ a,b ∈ S ( τ ( ab ) ⊆ τ a ∩ τ b ) ; (iv) aτ is an sd-convex subset of S and a ⊲⊳ aτ for every a ∈ S ; (v) τ a is an sd-ideal of S and a ⊲⊳ τ a for every a ∈ S .Proof. (i) ⇒ (iii), (v) ⇒ (i), (v) ⇒ (ii), (iv) ⇒ (ii). Those implications are proved in theTheorem 3.7.(iii) ⇒ (iv). By Lemma 3.3, aτ is an sd-subset of S such that a ⊲⊳ aτ for any a ∈ S .We have xy ∈ aτ ⇔ ( a, xy ) ∈ τ ⇔ a ∈ τ ( xy ) ⊆ τ x ∩ τ y ⇒ a ∈ τ x ∧ a ∈ τ y ⇔ x ∈ aτ ∧ y ∈ aτ. aτ is an sd-convex subset for any a ∈ S .(i) ⇒ (v). By Lemma 3.3, τ a is an sd-subset of S such that a ⊲⊳ τ a for any a ∈ S . Let a, x ∈ S be such that x ∈ τ a , i.e. ( x, a ) ∈ τ . By the co-transitivity of τ , we have( x, xs ) ∈ τ ∨ ( xs, a ) ∈ τ, for any s ∈ S . But, as τ is positive, we have only ( xs, a ) ∈ τ , i.e xs ∈ τ a . In the same wayone can prove that sx ∈ τ a . Thus, τ a is an ideal of S for any a ∈ S .(ii) ⇒ (v). By Lemma 3.3, τ a is an sd-subset of S , and a ⊲⊳ τ a for any a ∈ S . Now,let x ∈ τ a and s ∈ S . Then, by the co-transitivity of τ , we have ( x, xs ) ∈ τ or ( xs, a ) ∈ τ .If ( x, xs ) ∈ τ , then xs ∈ xτ ⊆ xτ ∪ sτ ⊆ ( xs ) τ , which is, by Lemma 3.3, impossible. Thus( xs, a ) ∈ τ . As ( sx, a ) ∈ τ can be proved similarly, we have proved that τ a is an sd-idealof S . Theorem 3.10.
Let τ be a complement positive co-quasiorder on a semigroup S . Thefollowing conditions are equivalent: (i) τ has the constructive cm-property; (ii) ∀ a,b ∈ S (( ab ) τ = aτ ∪ bτ ) ; (iii) τ a is an sd-completely isolated ideal of S such that a ⊲⊳ τ a for any a ∈ S .Proof. (i) ⇒ (ii). By Theorem 3.9, aτ ∪ bτ ⊆ ( ab ) τ for all a, b ∈ S . To prove the converseinclusion, take x ∈ ( ab ) τ . Then we have x ∈ ( ab ) τ ⇔ ( ab, x ) ∈ τ ⇒ ( a, x ) ∈ τ ∨ ( b, x ) ∈ τ by the lcm-property ⇔ x ∈ aτ ∨ x ∈ bτ ⇔ x ∈ aτ ∪ bτ. (ii) ⇒ (iii). By Theorem 3.9, τ a is an sd-ideal of S such that a ⊲⊳ τ a , for any a ∈ S .Let x, y ∈ S be such that xy ∈ τ a . Then a ∈ ( xy ) τ = xτ ∪ yτ by the assumption. Thus, a ∈ xτ or a ∈ yτ . So, x ∈ τ a or y ∈ τ a , and τ a is an sd-completely isolated ideal of S forany a ∈ S .(iii) ⇒ (i). Let a, b, c ∈ S be such that ( ab, c ) ∈ τ . Then, ab ∈ τ c and, since τ c iscompletely isolated, a ∈ τ c or b ∈ τ c , which means that ( a, c ) ∈ τ or ( b, c ) ∈ τ .Following Bishop, every classical theorem presents the challenge: find a constructiveversion with a constructive proof. This constructive version can be obtained by strength-ening the conditions or weakening the conclusion of the theorem. There are, often, severalconstructively different versions of the same classical theorem.Comparing the obtained results for complement positive co-quasiorders with the parallelones for positive quasiorders in the classical background, we can conclude that the classicalTheorem 2.1 breaks into two new ones in the constructive setting:32 Theorem 3.7 obtained by weakening the conclusions, • Theorem 3.9 obtained by strengthening the conditions - here strengthening the log-ical background. Recall that intermediate logic proves intuitionistically as well asclassically valid theorems, yet they often possess a strong constructive flavour.In addition, there are two definitions: those of the constructive cm-property, and thecomplement cm-property. Nevertheless, the last definition, Theorem 3.8, is stronger.
Remark 8.
For some classical theorems it is shown that they are not provable construc-tively. Some classical theorems are neither provable nor disprovable, that is, they areindependent of
BISH . Let us remember that in
CLASS the compatibility property is an important condition forproviding the semigroup structure on quotient sets. Now we are looking for the tools forintroducing an apartness relation on a factor semigroup. Our starting point is the resultsfrom Subsection 3.1.3, as well as the next definition.A co-equivalence κκκ is a co-congruence if it is co-compatible ∀ a,b,x,y ∈ S (( ax, by ) ∈ κ ⇒ ( a, b ) ∈ κ ∨ ( x, y ) ∈ κ ) Theorem 3.11.
Let S be a semigroup with apartness. Then (i) Let µ be a congruence, and κ a co-congruence on S . Then, κ defines an apartnesson the factor set S/µ if and only if µ ∩ κ = ∅ . (ii) The quotient mapping π : S → S/µ , defined by π ( x ) = xµ , is an onto se-homomorphism.Proof. (i). If κ defines an apartness on S/µ , then, by Theorem 3.2(i), µ ∩ κ = ∅ .Let µ be a congruence and κ a co-congruence on a semigroup with apartness S suchthat µ ∩ κ = ∅ . Then, by Theorem 3.2(i), κ defines apartness on S/µ .Let aµ xµ = bµ yµ , then ( ax ) µ ( by ) µ which further, by the definition of apartness on S/µ ,ensures that ( ax, by ) ∈ κ . But κ is a co-congruence, so either ( a, b ) ∈ κ or ( x, y ) ∈ κ . Thus,by the definition of apartness in S/µ again, either aµ bµ or xµ yµ . So ( S/µ, = , , · ) isa semigroup with apartness.(ii). By Theorem 3.2(ii), π is an onto se-mapping. By (i) and assumption, we have π ( xy ) = ( xy )( ∼ κ ) = x ( ∼ κ ) y ( ∼ κ ) = π ( x ) π ( y ) . Hence π is a homomorphism.As a consequence of Theorem 3.11 and Corollary 3.1 we have the next corollary. Corollary 3.3. If κ is a co-congruence on S , then the relation ∼ κ (= ¬ κ ) is a congruenceon S , and κ defines an apartness on S/ ∼ κ . he apartness isomorphism theorem for semigroups with apartness follows. Theorem 3.12.
Let f : S → T be an se-homomorphism between semigroups with apart-ness. Then: (i) coker f is a co-congruence on S , which defines an apartness on S/ ker f , (ii) the mapping θ : S/ ker f → T , defined by θ ( x (ker f )) = f ( x ) , is an apartness embed-ding such that f = θ ◦ π ; and (iii) if f maps S onto T , then θ is an apartness isomorphism.Proof. See [38].Recall, following [44],
BISH (and constructive mathematics in general) is not the studyof constructive things, it is a constructive study of things. In constructive proofs of classicaltheorems, only constructive methods are used.Although constructive theorems might look like the corresponding classical versions,they often have more complicated hypotheses and proofs. Comparing Theorem 2.4 (respec-tively Theorem 2.3) for classical semigroups and Theorem 3.12 (respectively Theorem 3.11)for semigroup with apartness, we have evidence for that.
During the implementation of the FTA Project [22], the notion of commutative construc-tive semigroups with tight apartness appeared. We put noncommutative constructivesemigroups with “ordinary” apartness in the centre of our study, proving first, of course,that such semigroups do exist. Once again we want to emphasize that semigroups withapartness are a new approach , and not a new class of semigroups.Let us give some examples of applications of ideas presented in the previous section.We will start with constructive analysis. The proof of one of the directions of the construc-tive version of the Spectral Mapping Theorem is based on some elementary constructivesemigroups with inequality techniques, [11]. It is also worth mentioning the applicationsof commutative basic algebraic structures with tight apartness within the automated rea-soning area, [15]. For possible applications within computational linguistic see [40]. Sometopics from mathematical economics can be approached constructively too (using someorder theory for sets with apartness), [2]. Contrary to the classical case, the applicationsof constructive semigroups with apartness, due to their novelty, constitute an unexploredarea. In what follows some possible connections between semigroups with apartness andcomputer science are sketched.semigroups with apartness semigroups with apartness l l bisimulation automated theorem proving l l l l process algebra artificial inteligence l transactions and concurrency l databasesOne of the directions of future work is to be able to say more about those links. The studyof basic constructive algebraic structures with apartness as well as constructive algebra asa whole can impact the development of other areas of constructive mathematics. On theother hand, it can make both proof engineering and programming more flexibile.Although the classical theory of semigroups has been considerably developed in thelast decades, constructive mathematics has not paid much attention to semigroup theory.One of our main scientific activities will be to further develop of the constructive theoryof semigroups with apartness. Semigroups will be examined constructively, that is withintuitionistic logic. To develop this constructive theory of semigroups with apartness, weneed first to clarify the notion of a set with apartness. The initial step towards groundingthe theory is done by our contributing papers [16], [17], [37], [38], [39] - a critical reviewof some of those results as well as the solutions to some of the open problems arising fromthose papers are presented in Section 3.Why should a mathematician choose to work in this manner? As it is written in oneof the reviews of Errett Bishop’s monograph Foundations of functional analysis , [49], “toreplace the classical system by the constructive one does not in any way mutilate the greatclassical theories of mathematics. Not at all. If anything, it strengthens them, and showsthem, in a truer light, to be far grander than we had known.” At heart, Bishop’s con-structive mathematics is simply mathematics done with intuitionistic logic, and may beregarded as “constructive mathematics for the working mathematician”, [51]. The mainactivity in the field consists in proving theorems rather than demonstrating the unprov-ability of theorems (or making other metamathematical observations), [3]. “Theorems aretools that make new and productive applications of mathematics possible,” [30].The theory of semigroups with apartness is, of course, in its infancy, but, as we havealready pointed out, it promises a prospective of applications in other (constructive) math-ematics disciplines, certain areas of computer science, social sciences, economics.To conclude, although one of the main motivators for initiating and developing thetheory of semigroups with apartness comes from the computer science area, in order to haveprofound applications, a certain amount of the theory, which can be applied, is necessaryfirst. Among priorities, besides the growing the general theory, are further developments of:constructive relational structures - (co)quotient structures in the first place, constructiveorder theory, theory of ordered semigroups with apartness, etc.The Summary of the European Commission’s
Mathematics for Europe , June 2016, [18],states that “mathematics should not only focus on nowadays’ applications but should leaveroom for development, even theoretical, that may be vital tomorrow.” With a strong belief35n the tomorrow’s vitalness of the theory of semigroups with apartness, the focus should beon its further development. On the other hand, it is useful to “leave room” for “nowadays’applications” as well. All those will represent the core of our forthcoming papers.
Acknowledgements
M. M. is supported by the Faculty of Mechanical Engineering, University of Niˇs, Serbia,Grant “Research and development of new generation machine systems in the function ofthe technological development of Serbia”. M. N. H. is supported by TWAS Research GrantRGA No. 17 - 542 RG / MATHS / AF / AC G -FR3240300147. The ICMPA-UNESCOChair is in partnership with Daniel Iagolnitzer Foundation (DIF), France, and the Associ-ation pour la Promotion Scientifique de l’Afrique (APSA), supporting the development ofmathematical physics in Africa.
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