aa r X i v : . [ m a t h . G R ] M a y ON GROUP-LIKE MAGMOIDS
DAN JONSSON
Abstract.
A magmoid is a non-empty set with a partial binary operation;group-like magmoids generalize group-like magmas such as groups, monoidsand semigroups. In this article, we first consider the many ways in which thenotions of associative multiplication, identities and inverses can be general-ized when the total binary operation is replaced by a partial binary operation.Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids,skew-prepoloids and skew-pregroupoids are then defined in terms of general-ized associativity, generalized identities and generalized inverses. Some basicresults about these magmoids are proved, and connections between poloid-likeand prepoloid-like magmoids, in particular semigroups, are derived. Notably,analogues of the Ehresmann-Schein-Nampooribad theorem are proved. Introduction
A binary operation m on a set S is usually defined as a mapping that assignssome m ( x, y ) ∈ S to every pair ( x, y ) ∈ S × S . Algebraists have been somewhatreluctant to work with partial functions, in particular partial binary operations,where m ( x, y ) is defined only for all ( x, y ) in some subset of S × S , often called thedomain of definition of m . For example, in the early 1950s Wagner [17] pointedout that if an empty partial transformation is interpreted as the empty relation ∅ then composition of partial transformations can be regarded as being defined forevery pair of transformations. Specifically, a partially defined binary system of non-empty partial transformations can be reduced to a semigroup which may containan empty partial transformation by interpreting composition of partial transforma-tions as composition of binary relations. This observation may have contributed tothe acceptance of the notion of a binary system of partial transformations [8]. Onthe other hand, the reformulation alleviated the need to come to terms with par-tial binary operations as such, thus possibly leaving significant research questionsunnoticed and unanswered.One kind of reason why algebraists have hesitated to embrace partial operationshas to do with the logical status of expressions such as f ( x ) = y and m ( x, y ) = z when f and m are partial functions. As Burmeister [1] explains, [a] first order language for algebraic systems is usually based on an appro-priate notion of equations. Such a notion [...] has already been aroundfor quite a long time, but approaches as by Kleene [...], Ebbinghaus [...],Markwald [...] and others (cf. also [Schein]) used then a three valuedlogic for the whole language (which might have deterred algebraists fromusing it). (pp. 306–7). One might argue that if ( x, y ) does not belong to the domain of definition of m then the assertion m ( x, y ) = z is meaningless, neither true nor false. This wouldseem to imply that we need some three-valued logic, where an assertion is notnecessarily either true or false, similar to Kleene’s three-valued logic [10]. Thisapproach is very problematic, however; one need only contemplate the meaningof an implication containing an assertion assumed to be neither true nor false toappreciate the complications that the use of a three-valued logic would entail. Fortunately, two-valued logic suffices to handle partial functions, including par-tial binary operations. In particular, three-valued logic is not needed if expressionsof the form f ( x , . . . , x n ) = y are used only if ( x , . . . , x n ) belongs to the domain ofdefinition of f . Burmeister [1] elaborated a formal logic with partial functions basedon this idea. Another, simpler way to stay within two-valued logic when dealingwith partial functions is sketched in Section 2 below.But there is also another important kind of reason why algebraists have shunnedpartial binary operations. Ljapin and Evseev [13] note that [it] often turns out that an idea embodying one clearly defined concept inthe theory of total operations corresponds to several mutually inequiva-lent notions in the theory of partial operations, each one reflecting oneor another aspect of the idea. (p. 17). For example, in a group-like system with a partial binary operation, multiplicationof elements can be associative in different ways, there are several types of identi-ties and inverses, and different kinds of subsystems and homomorphisms can bedistinguished. While this complicates algebraic theories using partial operations,it may be the case that these are differences that make a difference. Maybe theproliferation of notions just makes the theory richer and deeper, leading to a moreprofound understanding of the simpler special cases. Whether complexity equatesrichness and profoundness in this connection or not is a question that cannot reallybe answered a priori ; the answer must be based on experience from more or lesssuccessful use of partial operations in applications such as those in this article.In sum, there are no immediate reasons to avoid partial functions and opera-tions. Particularly in view of the fact that some systems with partially definedoperations, such as categories and groupoids, have received much attention formany years now, it would not be unreasonable to use a general theory of partialoperations as a foundation for a general theory of total operations. This has notyet happened in mainstream mathematics, however: the mainstream definition ofan algebra in Universal Algebra still uses total operations, not partially definedoperations. (While partial operations are not ignored, they are typically treatedas operations with special properties rather than as operations of the most generalform.) This article is a modest attempt to fill a little of the resulting void by gen-eralizing a theory employing total operations to a theory using partial operations.More concretely, we are concerned with “magmoids”: generalizations of magmas ob-tained by replacing the total binary operation by a partial operation. Specifically,“group-like” magmoids are considered; these generalize group-like magmas such assemigroups (without zeros), monoids and groups.Section 2 contains the definitions of magmoids and other fundamental concepts,and introduces a convenient notation applicable to partial binary operations andother partial mappings. It is also shown that the basic concepts can be defined ina way that does not lead to any logical difficulties. Sections 3 and 4 deal with themany ways in which the notions of associative multiplication, identities and inversesfrom group theory can be generalized when the total binary operation is replacedby a partial binary operation; some other concepts related to identities and inversesare considered in Section 5. In Sections 6 and 7 a taxonomy of group-like magmoidsbased on the distinctions presented in Section 4 is developed. Some results thatconnect the group-like magmoids in Section 6 to those in Section 7 are proved inSection 8.Much unconventional terminology is introduced in this article. This is becausethe core of the article is a systematic classification of some group-like magmoids,and the terminology reflects the logic of this classification. In some cases, thepresent terminology overlaps with traditional terminology, but the new terms are
N GROUP-LIKE MAGMOIDS 3 not meant to replace other, commonly used names of familiar concepts. Rather, thenaming scheme used here is intended to call attention to similarities and differencesbetween the notions distinguished.2.
Partial mappings and magmoids
Partial mappings.
Let X , . . . , X n be non-empty sets. An n-ary relation on X , . . . , X n , denoted r : X , . . . , X n , or just r when X , . . . , X n need not bespecified, is a tuple ( ρ, X , . . . , X n ) , where ρ ⊆ X × . . . × X n . The set ρ , also denoted γ r , is the graph of r . The emptyrelation on X , . . . , X n is the tuple ( ∅ , X , . . . , X n ) . The set { x i | ( x , . . . , x i , . . . , x n ) ∈ ρ } , a subset of X i denoted pr i ρ , is called the i:th projection of ρ . Note that if someprojection of ρ is the empty set then ρ itself is the empty set.A binary (2-ary) relation r : X, Y is thus a tuple ( ρ, X, Y ) such that ρ ⊆ X × Y . For r : X, Y we have pr ρ = { x | ( x, y ) ∈ ρ } and pr ρ = { y | ( x, y ) ∈ ρ } . We call pr ρ the effective domain of r , denoted edom r , and pr ρ the effective codomain of r , denoted ecod r . We also call X the total domain of r , denoted tdom r , and Y the total codomain of r , denoted tcod r . A total binaryrelation is a binary relation r such that edom r = tdom r , while a cototal binaryrelation is a binary relation r such that ecod r = tcod r . Definition 2.1. A functional relation, or (partial) mapping, f : X → Y is a binaryrelation ( φ, X, Y ) such that for each x ∈ pr φ there is exactly one y ∈ pr φ such that ( x, y ) ∈ φ . A self-mapping on X is a mapping f : X → X .We let f ( x ) = y express the fact that ( x, y ) ∈ γ f . Consistent with this, for any x ∈ edom f , f ( x ) denotes the unique element of Y such that ( x, f ( x )) ∈ γ f .Let f be a self-mapping on X . Then f ( x ) denotes some x ∈ X if and only if x ∈ edom f ; f ( f ( x )) denotes some x ∈ X if and only if x, f ( x ) ∈ edom f ; etc. Wedescribe such situations by saying that f ( x ) , f ( f ( x )) , etc. are defined .We let ( f ( x )) , ( f ( f ( x ))) etc. express the fact that f ( x ) , f ( f ( x )) etc. is defined. We also use this notation embedded in expressions, letting ( f ( x )) = y mean that f ( x ) is defined and f ( x ) = y , letting ( f ( x )) = ( g ( x )) mean that f ( x ) and g ( x ) aredefined and f ( x ) = g ( x ) , etc.It is important to note that if ( x, y ) ∈ X × Y but x / ∈ edom f = pr γ f then ( x, y ) / ∈ γ f , so if f ( x ) is not defined then f ( x ) = y is simply false, not meaningless.Also, f ( x ) = g ( x ) is equivalent to the condition that there is some y ∈ ecod f ∩ ecod g such that f ( x ) = y and g ( x ) = y , so such expressions do not present any newlogical difficulties, although expressions such as ( f ( x )) = ( g ( x )) generally describesituations of more interest. One could extend the scope of this notation, letting ( x ) , ( x ′ ) , . . . mean that x, x ′ , . . . belong to X , but in this article I will adhere to the more familiar, light-weight notation when dealing with’naked’ variables, writing ( φ ( x )) = y rather than ( φ ( x )) = ( x ′ ) or ( φ ( x )) = ( y ) , etc. DAN JONSSON
Binary operations and magmoids.Definition 2.2. A (partial) binary operation on a non-empty set X is a non-empty(partial) mapping m : X × X → X, ( x, y ) m ( x, y ) =: xy. A total binary operation on X is a total mapping m : X × X → X .A magmoid is a non-empty set X equipped with a (partial) binary operation on X ; a total magmoid, or magma, is a non-empty set X equipped with a total binaryoperation on X .Recall that in Definition 2.1, we identified a (partial) mapping f : X → Y witha binary relation ( φ, X, Y ) such that ( x, y ) , ( x, y ′ ) ∈ φ implies y = y ′ . We cansimilarly identify a (partial) binary operation m : X × X → X with a ternaryrelation ( µ, X, X, X ) such that ( x, y, z ) , ( x, y, z ′ ) ∈ µ implies z = z ′ , letting m ( x, y ) = z mean that ( x, y, z ) ∈ µ . In this case, we have edom m = { ( x, y ) | ( x, y, z ) ∈ µ } , tdom m = X × X , ecod m = { z | ( x, y, z ) ∈ µ } and tcod m = X .The notion of being defined for expressions involving a self-mapping can beextended in a natural way to expressions involving a binary operation. We say that xy is defined if and only if ( x, y ) ∈ edom m ; that ( xy ) z is defined if and only if ( x, y ) , ( xy, z ) ∈ edom m ; that z ( xy ) is defined if and only if ( x, y ) , ( z, xy ) ∈ edom m ;and so on. Thus, if ( xy ) z or z ( xy ) is defined then xy is defined. In analogywith the notation ( f ( x )) , we let ( xy ) mean that xy is defined, (( xy ) z ) mean that ( xy ) z is defined, ( x ( yz )) mean that x ( yz ) is defined, etc. Note that there is noconflict between the usual function of parentheses, namely to specify priority ofoperations, and their additional use here to show that a function is defined for acertain argument.It is clear that ( x, y ) / ∈ edom m implies m ( x, y ) = z for every z ∈ X , so in thiscase, too, we do not have to deal with logical anomalies. Remark . We have implicitly used lazy evaluation of conjunctions in this section.That is, the conjunction p ∧ q is evaluated step-by-step according to the followingalgorithm: if p is false then p ∧ q is false;else if q is false then p ∧ q is false;else p ∧ q is true . For example, if x / ∈ edom f then f ( f ( x )) is not defined; if x ∈ edom f but instead f ( x ) / ∈ edom f then f ( f ( x )) is also not defined; otherwise, f ( f ( x )) is defined. Thus,the question if f ( f ( x )) is defined does not arise before we know if f ( x ) is defined. Remark.
Burmeister [1] added some new primitives to standard logic to handle par-tially defined functions, and similarly the present approach ultimately requires aslight modification of standard logic, namely in the interpretation of conjunctions.However, it is important to note that the dynamic (lazy) evaluation interpreta-tion of conjunctions is fully consistent with the static standard interpretation ofconjunctions in terms of truth tables.3.
Conditions used in basic definitions
A group is a magma where multiplication is associative, and where there is anidentity element and an inverse for every element. In this section, we distinguishcomponents of these three properties of groups as they apply to magmoids.
N GROUP-LIKE MAGMOIDS 5
Associativity.
In a magma M , an associative binary operation is one thatsatisfies the condition x ( yz ) = ( xy ) z for all x, y, z ∈ M . If the magma is regardedas a magmoid P , we write this as(TA) ( x ( yz )) = (( xy ) z ) for all x, y, z ∈ P .In a magmoid we can in addition define conditions that generalize (TA):(A1) If ( x ( yz )) then ( x ( yz )) = (( xy ) z ) for all x, y, z ∈ P .(A2) If (( xy ) z ) then ( x ( yz )) = (( xy ) z ) for all x, y, z ∈ P .(A3) If ( xy ) and ( yz ) then ( x ( yz )) = (( xy ) z ) for all x, y, z ∈ P .These elementary conditions concern different aspects of associativity, and can beused as building blocks for constructing more complex conditions; see Section 4.1.In semigroups, we can omit parentheses, writing x ( yz ) and ( xy ) z as xyz withoutambiguity. In fact, it can be shown by induction using x ( yz ) = ( xy ) z that wecan write x · · · x n for any n without ambiguity; this is the so-called law of generalassociativity. Similarly, in a magmoid where (A1) and (A2) hold we can write ( x · · · x n ) without ambiguity when all subproducts are defined, so we have a generalassociativity law in this case, too.Specifically, let π x ,...,x n denote any parenthesized products of x , . . . , x n , in thisorder, for example, ( x ( · · · )) or (( · · · ) x n ) . If (A1) and (A2) hold and π ′ x ,...,x n is aparenthesized product of the same kind then it can be shown by induction that π x ,...,x n = π ′ x ,...,x n . For example, if ( x ( y ( zu ))) then ( x ( y ( zu ))) = ( x (( yz ) u )) = (( x ( yz )) u ) = ((( xy ) z ) u ) = (( xy )( zu )) . In other words, π x ,...,x n is uniquely determined by the sequence x , . . . , x n , so wecan write any π x ,...,x n as ( x · · · x n ) . For example, we can write (( xy )( zu )) as ( xyzu ) without loss of information.(A3) thus implies that if ( xy ) and ( yz ) then ( x ( yz )) = (( xy ) z ) = ( xyz ) . Byrepeatedly applying (A1) and (A2) together with (A3), we can generalize (A3);for example, if ( xyz ) and ( zuv ) then ( xyzuv ) . We can also generalize (A3) byapplying it repeatedly; for example, if ( xy ) , ( yz ) and ( zu ) then ( xyzu ) . Combiningthese two ways of generalizing (A3), it becomes possible to make inferences such as” if ( x x ) and ( x x x x ) and ( x x x ) then ( x x x x x x x ) ”.Note that we can retain certain redundant inner parentheses for emphasis. Forexample, if (cid:0) xx − x (cid:1) = x then we can write (cid:0)(cid:0) xx − x (cid:1) y (cid:1) = ( xy ) instead of (cid:0) xx − xy (cid:1) = ( xy ) to clarify how the equality is established.3.2. Identities.
In a magma M , an identity is an element e ∈ M such that ex = xe = x for all x ∈ M . More generally, a left (resp. right) identity is an element e ∈ M such that ex = x (resp. xe = x ) for all x ∈ M . In a magmoid P , theconditions defining left and right identities take the following forms:(TU1) If x ∈ P then ( ex ) = x .(TU2) If x ∈ P then ( xe ) = x .In a magmoid we can in addition define conditions generalizing (TU1) and (TU2):(GU1) If x ∈ P and ( ex ) then ( ex ) = x .(GU2) If x ∈ P and ( xe ) then ( xe ) = x .(LU1) There is some x ∈ P such that ( ex ) = x .(LU2) There is some x ∈ P such that ( xe ) = x .In Section 4.2, we define different types of identities (units) in magmoids by meansof combinations of these conditions. DAN JONSSON
Inverses.
In a magma M , an inverse of x ∈ M is an element x − ∈ M suchthat xx − = x − x = e for some identity e ∈ M , while a right (resp. left) inverseof x ∈ M is an element x − ∈ M such that xx − = e (resp. x − x = e ) for someidentity e ∈ M . We can also define a right (resp. left) semi-inverse of x ∈ M asan element x − ∈ M such that xx − = e (resp. x − x = e ) for some left or rightidentity e ∈ M ; these are the most fundamental notions. In a magmoid P , theconditions defining a left or right semi-inverse x − take the following forms:(TI1) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (TU1).(TI2) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (TU1).(TI3) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (TU2).(TI4) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (TU2).In addition, we can define conditions that generalize (TI1) – (TI4):(GI1) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (GU1).(GI2) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (GU1).(GI3) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (GU2).(GI4) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (GU2).(LI1) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (LU1).(LI2) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (LU1).(LI3) If x ∈ P then (cid:0) xx − (cid:1) = e for some e satisfying (LU2).(LI4) If x ∈ P then (cid:0) x − x (cid:1) = e for some e satisfying (LU2).In Section 4.3, different types of inverses in magmoids will be defined in terms ofconditions of this kind. 4. Basic definitions
Some of the possible combinations of elementary conditions in Section 3 will beused in this section to define different types of associatitvity, units (identities) andinverses. There are two main themes in this section: one-sided versus two-sidednotions, and local versus global notions.4.1.
Magmoids according to types of associativity.Definition 4.1.
Let P be a magmoid, x, y, z ∈ P . Consider the following con-ditions:(S1) If ( x ( yz )) or if ( xy ) and ( yz ) then ( x ( yz )) = (( xy ) z ) .(S2) If (( xy ) z ) or if ( xy ) and ( yz ) then ( x ( yz )) = (( xy ) z ) .(S3) If ( x ( yz )) or (( xy ) z ) or if ( xy ) and ( yz ) then ( x ( yz )) = (( xy ) z ) .A left semigroupoid is a magmoid satisfying (S1), a right semigroupoid is a magmoidsatisfying (S2), and a (two-sided) semigroupoid is a magmoid satisfying (S3).It is clear that if P is a magma and at least one of the conditions (S1) – (S3) issatisfied then P is a semigroup; conversely, if P is a semigroup then P is a magmawhere (S1) – (S3) are trivial implications so that they are all satisfied.In view of the symmetry between left and right semigroupoids essentially onlytwo cases will be considered below, namely left (or right) semigroupoids and (two-sided) semigroupoids. For later use, we note that a left semigroupoid is a magmoidsuch that for all x, y, z ∈ P (4.1) ( x ( yz )) → ( x ( yz )) = (( xy ) z ) , (( xy ) ∧ ( yz )) → ( x ( yz )) = (( xy ) z ) , while a semigroupoid is a left semigroupoid such that in addition to (4.1) we have(4.2) (( xy ) z ) → ( x ( yz )) = (( xy ) z ) . N GROUP-LIKE MAGMOIDS 7
Types of units in magmoids.Definition 4.2.
Let P be a magmoid, x ∈ P .(1) A (global) left unit is some ǫ ∈ P such that if ( ǫx ) then ( ǫx ) = x , while a (global) right unit is some ε ∈ P such that if ( xε ) then ( xε ) = x .(2) A (global) two-sided unit is some e ∈ P which is a (global) left unit and a(global) right unit.We denote the set of left units, right units, and two-sided units in P by { ǫ } P , { ε } P , and { e } P , respectively. Definition 4.3.
Let P be a magmoid, x ∈ P .(1) A local left unit for x is some λ x ∈ P such that ( λ x x ) = x ; a local rightunit for x is some ρ x ∈ P such that x = ( xρ x ) .(2) A twisted left unit ϕ x ∈ P for x is a local left unit for x which is a rightunit, while a twisted right unit ψ x ∈ P for x is a local right unit for x whichis a left unit.(3) A left effective unit ℓ x ∈ P for x is a local left unit for x which is a two-sided unit, while a right effective unit r x ∈ P for x is a local right unit for x which is a two-sided unit.We denote the set of local left (resp. local right) units for x ∈ P by { λ } x (resp. { ρ } x ), the set of twisted left (resp. twisted right) units for x ∈ P by { ϕ } x (resp. { ψ } x ), and the set of left effective (resp. right effective) units for x ∈ P by { ℓ } x (resp. { r } x ). We also set { λ } P = ∪ x ∈ P { λ } x , { ρ } P = ∪ x ∈ P { ρ } x , { ϕ } P = ∪ x ∈ P { ϕ } x , { ψ } P = ∪ x ∈ P { ψ } x , { ℓ } P = ∪ x ∈ P { ℓ } x and { r } P = ∪ x ∈ P { r } x .4.3. Types of inverses in magmoids.Definition 4.4.
Let P be a magmoid, x ∈ P .(1) A pseudoinverse of x is some x ( − ∈ P such that (cid:0) xx ( − (cid:1) ∈ { λ } x and (cid:0) x ( − x (cid:1) ∈ { ρ } x .(2) A right preinverse of x is some x − ∈ P such that (cid:0) xx − (cid:1) ∈ { λ } x and (cid:0) x − x (cid:1) ∈ { λ } x − .(3) A left preinverse of x is some x − ∈ P such that (cid:0) x − x (cid:1) ∈ { ρ } x and (cid:0) xx − (cid:1) ∈ { ρ } x − .(4) A (two-sided) preinverse of x is some x − ∈ P such that we have (cid:0) xx − (cid:1) ∈ { λ } x ∩ { ρ } x − and (cid:0) x − x (cid:1) ∈ { ρ } x ∩ { λ } x − .Thus, x ( − is a pseudoinverse of x if and only if (cid:0)(cid:0) xx ( − (cid:1) x (cid:1) = (cid:0) x (cid:0) x ( − x (cid:1)(cid:1) = x , x − is a right preinverse of x if and only if (cid:0)(cid:0) xx − (cid:1) x (cid:1) = x and (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = x − , x − is a left preinverse of x if and only if (cid:0) x (cid:0) x − x (cid:1)(cid:1) = x and (cid:0) x − (cid:0) xx − (cid:1)(cid:1) = x − ,and x − is a preinverse of x if and only if (cid:0)(cid:0) xx − (cid:1) x (cid:1) = (cid:0) x (cid:0) x − x (cid:1)(cid:1) = x and (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = (cid:0) x − (cid:0) xx − (cid:1)(cid:1) = x − .Let J , I + , I ∗ and I be binary relations on a magmoid P such that x J x if andonly if (( xx ) x ) = ( x ( xx )) = x , x I + x if and only if (( xx ) x ) = x and (( xx ) x ) = x , x I ∗ x if and only if ( x ( xx )) = x and ( x ( xx )) = x , and x I x if and only if x I + x and x I ∗ x . In terms of these relations, x is a pseudoinverse of x if and only if x J x , aright preinverse of x if and only if x I + x , a left preinverse of x if and only if x I ∗ x ,and a preinverse of x if and only if x I x . Note that I + , I ∗ and I are symmetricrelations.It is useful to have some special notation for sets of pseudoinverses and prein-verses, and we set J {} x = { x | x J x } , I + {} x = { x | x I + x } , I ∗ {} x = { x | x I ∗ x } and I {} x = { x | x I x } . DAN JONSSON
Definition 4.5.
Let P be a magmoid, x ∈ P .(1) A strong pseudoinverse of x is a pseudoinverse x ( − of x such that (cid:0) xx ( − (cid:1) , (cid:0) x ( − x (cid:1) ∈ { e } P .(2) A strong right preinverse of x is a right preinverse x − of x such that (cid:0) xx − (cid:1) , (cid:0) x − x (cid:1) ∈ { ε } P .(3) A strong left preinverse of x is a left preinverse x − of x such that (cid:0) x − x (cid:1) , (cid:0) xx − (cid:1) ∈ { ǫ } P .(4) A strong (two-sided) preinverse of x is a preinverse x − of x such that (cid:0) xx − (cid:1) , (cid:0) x − x (cid:1) ∈ { e } P .By this definition, a strong right preinverse of x is a right preinverse x − of x such that (cid:0) y (cid:0) xx − (cid:1)(cid:1) = (cid:0) y (cid:0) x − x (cid:1)(cid:1) = y for all y ∈ P , and a strong preinverseof x is a preinverse x − of x such that (cid:0) y (cid:0) xx − (cid:1)(cid:1) = (cid:0) y (cid:0) x − x (cid:1)(cid:1) = (cid:0)(cid:0) xx − (cid:1) y (cid:1) = (cid:0)(cid:0) x − x (cid:1) y (cid:1) = y for all y ∈ P . Strong left preinverses and strong pseudoinverseshave corresponding properties.We let x J x , x I + x , x I ∗ x and x I x mean that x is a strong pseudoinverse,strong right preinverse, strong left preinverse and strong preinverse of x , respec-tively. It is clear that I + , I ∗ and I are symmetrical relations. We also set J {} x = { x | x J x } , I + {} x = (cid:8) x | x I + x (cid:9) , I ∗ {} x = { x | x I ∗ x } and I {} x = { x | x I x } .4.4. Canonical local units.
Recall that if x − is a left, right or two-sided pre-inverse of x then (cid:0) xx − (cid:1) ∈ { λ } x ∪ { ρ } x − and (cid:0) x − x (cid:1) ∈ { ρ } x ∪ { λ } x − . We set λ x = ρ x − = (cid:0) xx − (cid:1) , ρ x = λ x − = (cid:0) x − x (cid:1) , and call such local units canonical localunits. We may also use the notation { λ } x (resp. { ρ } x ) for the set of canonical localleft (resp. right) units for x , and the notation { λ } x − (resp. { ρ } x − ) for the set ofcanonical local left (resp. right) units for x − . We also set { λ } P = ∪ x ∈ P { λ } x and { ρ } P = ∪ x ∈ P { ρ } x .A canonical local unit of the form (cid:0) xx . − (cid:1) (resp. (cid:0) x − x (cid:1) ) such that ( xx ′ ) = ( xx ′′ ) (resp. ( x ′ x ) = ( x ′′ x ) ) for any inverses x ′ , x ′′ of x is said to be unique .If we regard x − not as a preinverse of x ∈ P but just as an element x − ∈ P ,we write λ ( x − ) instead of λ x − and ρ ( x − ) instead of ρ x − , setting λ ( x − ) = (cid:16) x − (cid:0) x − (cid:1) − (cid:17) , ρ ( x − ) = (cid:16)(cid:0) x − (cid:1) − x − (cid:17) for some preinverse (cid:0) x − (cid:1) − of x − . Note, though, that if x I x − then x − I x ,so x ∈ I {} x − , so if λ ( x − ) is unique then (cid:0) x − x (cid:1) = (cid:16) x − (cid:0) x − (cid:1) − (cid:17) , meaning that λ x − = λ ( x − ) ; similarly, if ρ ( x − ) is unique then ρ x − = ρ ( x − ) . These identitieshold for left and right preinverses as well, since I + and I ∗ are symmetric relations. Remark . It is natural to write y = λ x when { y } = { λ } x , y = ρ x when { y } = { ρ } x , y = x − when { y } = I + {} x , { y } = I ∗ {} x , { y } = I {} x , { y } = I + {} x , { y } = I ∗ {} x or { y } = I {} x , and so on.5. Idempotents and involution
Idempotents in magmoids.Definition 5.1. An idempotent in a magmoid P is some i ∈ P such that ( ii ) = i . Proposition 5.1.
Let P be a magmoid, i ∈ P . If ( ii ) = i then i ∈ { λ } i ∩ { ρ } i .Proof. If ( ii ) = i then i ∈ { λ } i and i ∈ { ρ } i . (cid:3) Corollary 5.1.
Let P be a magmoid with unique local units, i ∈ P . If ( ii ) = i then i = λ i = ρ i . N GROUP-LIKE MAGMOIDS 9
Proposition 5.2.
Let P be a magmoid, i ∈ P . If ( ii ) = i then i ∈ I + {} i , i ∈ I ∗ {} i and i ∈ I {} i .Proof. If ( ii ) = i then (( ii ) i ) = ( i ( ii )) = i , so i I + i , i I ∗ i and i I i . (cid:3) Corollary 5.2.
Let P be a magmoid where right preinverses, left preinverses or(two-sided) preinverses are unique, i ∈ P . If ( ii ) = i then i − = i .Proof. We have i ∈ I + {} i = (cid:8) i − (cid:9) or i ∈ I ∗ {} i = (cid:8) i − (cid:9) or i ∈ I {} i = (cid:8) i − (cid:9) . (cid:3) Proposition 5.3.
Let P be a left (resp. right) semigroupoid with unique right(resp. left) preinverses, or a semigroupoid with unique preinverses. Then (cid:0) xx − (cid:1) and (cid:0) x − x (cid:1) are idempotents for every x ∈ P , and for every idempotent i ∈ P thereis some x ∈ P such that i = (cid:0) xx − (cid:1) = (cid:0) x − x (cid:1) .Proof. It suffices two consider the first two cases. In both of these, (cid:0) xx − (cid:1) and (cid:0) x − x (cid:1) so that (cid:0) x (cid:0) x − (cid:0) xx − (cid:1)(cid:1)(cid:1) , (cid:0) x − (cid:0) x (cid:0) x − x (cid:1)(cid:1)(cid:1) , (cid:0)(cid:0)(cid:0) x − x (cid:1) x − (cid:1) x (cid:1) and (cid:0)(cid:0)(cid:0) xx − (cid:1) x (cid:1) x − (cid:1) , and in a left (resp. right) semigroupoid we have (cid:0) x (cid:0) x − (cid:0) xx − (cid:1)(cid:1)(cid:1) = (cid:0)(cid:0) xx − (cid:1)(cid:0) xx − (cid:1)(cid:1) = (cid:0)(cid:0)(cid:0) xx − (cid:1) x (cid:1) x − (cid:1) = (cid:0) xx − (cid:1) , (cid:0) x − (cid:0) x (cid:0) x − x (cid:1)(cid:1)(cid:1) = (cid:0)(cid:0) x − x (cid:1)(cid:0) x − x (cid:1)(cid:1) = (cid:0)(cid:0)(cid:0) x − x (cid:1) x − (cid:1) x (cid:1) = (cid:0) x − x (cid:1) , (resp . (cid:0)(cid:0)(cid:0) x − x (cid:1) x − (cid:1) x (cid:1) = (cid:0)(cid:0) x − x (cid:1)(cid:0) x − x (cid:1)(cid:1) = (cid:0) x − (cid:0) x (cid:0) x − x (cid:1)(cid:1)(cid:1) = (cid:0) x − x (cid:1) , (cid:0)(cid:0)(cid:0) xx − (cid:1) x (cid:1) x − (cid:1) = (cid:0)(cid:0) xx − (cid:1)(cid:0) xx − (cid:1)(cid:1) = (cid:0) x (cid:0) x − (cid:0) xx − (cid:1)(cid:1)(cid:1) = (cid:0) xx − (cid:1) ) . Conversely, if i ∈ P is an idempotent then i = i − by Corollary 5.2, so i = ( ii ) = (cid:0) ii − (cid:1) = (cid:0) i − i (cid:1) . (cid:3) Involution magmoids.
A magma M may be equipped with a total self-mapping ∗ : x x ∗ such that ( x ∗ ) ∗ = x and ( xy ) ∗ = y ∗ x ∗ . A total self-mappingwith these properties is an anti-endomorphism on M by the second condition, and abijection by the first condition, so ∗ is an anti-automorphism, called an involutionfor M . This notion can be generalized to magmoids.
Definition 5.2. A (total) involution magmoid is a magmoid P equipped with atotal mapping ∗ : P → P, x
7→ ∗ ( x ) =: x ∗ such that ( x ∗ ) ∗ = x and if ( xy ) then ( xy ) ∗ = ( y ∗ x ∗ ) for all x, y ∈ P . We call thefunction x x ∗ an involution and x ∗ the involute of x .Involutes are inverse-like elements, but while inverses are defined in terms ofunits of various kinds (as elaborated in Sections 3.3 and 4.3), involutes are definedwithout reference to units, so involutes generalize inverses to situations where unitsmay not be available. Conversely, however, unit-like elements may be defined interms of involutes. Definition 5.3. A unity in a magmoid P with involution ∗ is some u ∈ P suchthat u ∗ = u .For any x ∈ P , ( xx ∗ ) and ( x ∗ x ) are unities since ( xx ∗ ) ∗ = (cid:0) ( x ∗ ) ∗ x ∗ (cid:1) = ( xx ∗ ) and ( x ∗ x ) ∗ = (cid:0) x ∗ ( x ∗ ) ∗ (cid:1) = ( x ∗ x ) . If P is a semigroupoid and ( xx ∗ x ) = x then ( x ∗ xx ∗ x ) = ( x ∗ x ) and ( xx ∗ xx ∗ ) = ( xx ∗ ) , so that ( xx ∗ ) and ( x ∗ x ) are idempotents. If x ∗ = x ∗ then x = (cid:0) x ∗ (cid:1) ∗ = (cid:0) x ∗ (cid:1) ∗ = x , and if x ∈ M then x = ( x ∗ ) ∗ ∈ ( M ∗ ) ∗ ⊆ M ∗ since M ∗ ⊆ M , so M ⊆ M ∗ , so M = M ∗ . Hence, x x ∗ is injective and surjective. This is the definition in semigroup theory; in general mathematics an involution is usuallydefined as a self-mapping ∗ such that ( x ∗ ) ∗ = x . We can use unities to define a kind of inverses, just as we used units to defineinverses in Section 4.3. Let P be an involution magmoid, x ∈ P . An involutionpseudoinverse of x is some x (+) ∈ P such that (cid:16)(cid:16) xx (+) (cid:17) x (cid:17) = (cid:16) x (cid:16) x (+) x (cid:17)(cid:17) = x, where (cid:0) xx (+) (cid:1) and (cid:0) x (+) x (cid:1) are unities, while an involution preinverse of x is some x + ∈ P such that (cid:0)(cid:0) xx + (cid:1) x (cid:1) = (cid:0) x (cid:0) x + x (cid:1)(cid:1) = x, (cid:0)(cid:0) x + x (cid:1) x + (cid:1) = (cid:0) x + (cid:0) xx + (cid:1)(cid:1) = x + , where ( xx + ) and ( x + x ) are unities. It is easy to show that in a semigroupoid thereis at most one involution preinverse for each x ∈ P .In the semigroupoid of all matrices over R or C , where ( M N ) if and only if M is a p × q matrix and N is a q × r matrix, the transpose A T and conjugatetranspose A † of a p × q matrix A are involutes of A . The unities are then symmetricor Hermitian matrices, that is, matrices such that A T = A or A † = A . Theinvolution preinverse of A is the unique q × p matrix A + such that ( AA + A ) = A , ( A + AA + ) = A + and such that the p × p matrix ( AA + ) and the q × q matrix ( A + A ) are unities, that is, symmetric or Hermitian matrices for which ( AA + ) ∗ = ( AA + ) and ( A + A ) ∗ = ( A + A ) . A + is known as the Moore-Penrose inverse of A .Total involutions can be generalized to partial involutions; this notion is alsoof interest. For example, it is easy to verify that the partial function A A − ,which associates every invertible matrix with its inverse, is a partial involution inthe semigroupoid of matrices over R or C . In every subsemigroupoid of invertible n × n matrices over R or C , A A − is a total involution, and A − is an involutionpreinverse as well as an involute, so A + = A − since involution preinverses areunique – recall that the Moore-Penrose inverse generalizes the ordinary matrixinverse.The relationship between involution magmoids and corresponding semiheapoidsis briefly described in Appendix A.6. Prepoloids and related magmoids
The magmoids considered in this section are equipped with local units. In theliterature, the focus is on the special case when these magmoids are magmas, namelysemigroups, but here we generalize such magmas to magmoids.6.1.
The prepoloid family.Definition 6.1.
Let P be a semigroupoid, Then P is(1) a prepoloid when there is a local left unit λ x ∈ P and a local right unit ρ x ∈ P for every x ∈ P ; (2) a pregroupoid when P is a prepoloid such that for every x ∈ P there is apreinverse x − ∈ P of x .This means that a prepoloid P is a semigroupoid such that for every x ∈ P thereare local units λ x , ρ x such that ( λ x x ) = ( xρ x ) = x . A pregroupoid is a prepoloid P such that for every x ∈ P there is some x − ∈ P such that (cid:0) xx − (cid:1) ∈ { λ } x ∩ { ρ } x − and (cid:0) x − x (cid:1) ∈ { ρ } x ∩ { λ } x − . If α ji = a ij , β kj = b jk and γ ki = P j a ij b jk then γ ki = P j β kj α ji and γ ki = P j β kj α ji . Involution inverses are unique when they exist, and it can be shown that every matrix A hasan involution inverse A + with respect to the involutions A A T and A A † . ’Prepoloids’ generalize the ’poloids’ considered in Section 7. The term ’poloid’ for a generalizedmonoid was introduced and motivated in [9]. Kock [11] lets the term ’pregroupoid’ refer to a set with a partially defined ternary operation.
N GROUP-LIKE MAGMOIDS 11
Prepoloids.
Proposition 6.1.
Let P be a prepoloid, x, y ∈ P , ρ x ∈ { ρ } x and λ y ∈ { λ } y . If ρ x = λ y then ( xy ) .Proof. If ρ x = λ y then ( ρ x y ) since ( λ y y ) , so (( xρ x ) y ) = ( xy ) since ( xρ x ) = x . (cid:3) Proposition 6.2.
Let P be a prepoloid with unique local units, x ∈ P . Then ( λ x λ x ) = λ x = λ λ x = ρ λ x and ( ρ x ρ x ) = ρ x = ρ ρ x = λ ρ x .Proof. If x ∈ P then x = ( λ x x ) = ( λ x ( λ x x )) = (( λ x λ x ) x ) , so ( λ x λ x ) ∈ { λ } x = { λ x } . Also, ( λ x λ x ) = λ x implies that λ x ∈ { λ } λ x = { λ λ x } and λ x ∈ { ρ } λ x = { ρ λ x } .It is proved similarly that ( ρ x ρ x ) = ρ x = ρ ρ x = λ ρ x . (cid:3) By Propositions 5.1 and 6.2, an element of a prepoloid with unique local unitsis thus a local unit if and only if it is an idempotent.
Proposition 6.3.
Let P be a prepoloid with unique local units, x, y ∈ P . If ( xy ) then λ ( xy ) = λ x and ρ ( xy ) = ρ y .Proof. If ( xy ) then ( xy ) = (( λ x x ) y ) = ( λ x ( xy )) , so λ x ∈ { λ } ( xy ) = (cid:8) λ ( xy ) (cid:9) . Simi-larly, ( xy ) = ( x ( yρ y )) = (( xy ) ρ y ) , so ρ y ∈ { ρ } ( xy ) = (cid:8) ρ ( xy ) (cid:9) . (cid:3) A prepoloid P with unique local units can be equipped with unique surjectivefunctions s : P → { λ } P , x λ x , t : P → { ρ } P , x ρ x , such that(6.1) ( s ( x ) x ) = x, ( x t ( x )) = x. By Proposition 6.2, s ( λ x ) = λ x for all λ x ∈ { λ } P and t ( ρ x ) = ρ x for all ρ x ∈ { ρ } P .A prepoloid with unique local units can thus be regarded as a semigroupoid ( P, m ) expanded to a prepoloid ( P, m , s , t ) characterized by the uniqueness property andthe identities (4.1), (4.2), and (6.1).Note that there may exist more than one function s : P → { λ } P such that ( s ( x ) x ) = x for all x ∈ P , and more than one function t : P → { ρ } P such that ( x t ( x )) = x for all x ∈ P (see Example 6.1).We call a prepoloid which admits not necessarily unique functions s , t : P → P satisfying (6.1) a bi-unital prepoloid ; those bi-unital prepoloids which are magmasare bi-unital semigroups. These are thus characterized by the identities(6.2) x ( yx ) = ( xy ) z, s ( x ) x = x, x t ( x ) = x. The class of bi-unital semigroups includes many types of semigroups studied inthe literature such as the function systems of Schweitzer and Sklar [15], abundantsemigroups [4], adequate semigroups [3], Ehresmann semigroups [12], ample semi-groups and restriction semigroups (see, e.g., [7]), and regular and inverse semigroupswith s and t defined by s ( x ) = (cid:0) xx − (cid:1) and t ( x ) = (cid:0) x − x (cid:1) .As we have seen, several identities, such as s ( s ( x )) = s ( x ) , ( s ( x ) s ( x )) = s ( x ) and s ( xy ) = s ( x ) , can be derived from the assumption that local units are unique.Bi-unital pregroupoids and semigroups where this is not postulated can be requiredto satisfy other conditions in order to have desirable properties; various such re-quirements are used to characterize the bi-unital semigroups found in the literature. Pregroupoids (1).
It is a well-known result in semigroup theory that a semigroup S has a pseudoinverse x ( − such that xx ( − x = x for every x ∈ S if and only if S has a so-called generalized inverse x − such that xx − x = x and x − xx − = x − for every x ∈ S . A regular semigroup can thus be defined by either condition. Thisresult can be generalized to semigroupoids. Proposition 6.4.
Let P be a semigroupoid. Each x ∈ P has a preinverse x − ∈ P if and only if each x ∈ P has a pseudoinverse x ( − ∈ P .Proof. Trivially, each preinverse x − of x is a pseudoinverse x ( − of x . Conversely,if x is a pseudoinverse of x then ( xxx ) , ( xx ) and ( xx ) . Thus, ( xxx ) , ( xxxxx ) and ( xxxxxxx ) , and as ( xxx ) = x we have ( x ( xxx ) x ) = ( xxxxx ) = ( xxx ) = x, (( xxx ) x ( xxx )) = ( xxxxxxx ) = ( xxxxx ) = ( xxx ) , so x I ( xxx ) , meaning that ( xxx ) ∈ I {} x . (cid:3) Recall that an inverse semigroup can be defined as a regular semigroup whosepreinverses are unique, or as a regular semigroup whose idempotents commute.These two characterizations are equivalent also when inverse semigroups are gen-eralized to pregroupoids.
Lemma 6.1.
Let P be a pregroupoid with unique preinverses, i, j ∈ P . If i, j areidempotents and ( ij ) then ( ij ) and ( ji ) are idempotents.Proof. We have (cid:16) ( ij )( ij ) − ( ij ) (cid:17) , so (cid:16) ij ( ij ) − ij (cid:17) , so (cid:16) j ( ij ) − i (cid:17) . Since ( ii ) , ( jj ) and ( ij ) , (cid:16) j ( ij ) − i (cid:17) implies (cid:16) ijj ( ij ) − i (cid:17) and (cid:16) j ( ij ) − iij (cid:17) . Thus, (cid:16) ijj ( ij ) − iij (cid:17) and (cid:16) j ( ij ) − iijj ( ij ) − i (cid:17) , and we have (cid:16)(cid:16) j ( ij ) − i (cid:17) ( ij ) (cid:16) j ( ij ) − i (cid:17)(cid:17) = (cid:16)(cid:16) j ( ij ) − i (cid:17)(cid:16) j ( ij ) − i (cid:17)(cid:17) = (cid:16) j (cid:16) ( ij ) − ( ij )( ij ) − (cid:17) i (cid:17) = (cid:16) j ( ij ) − i (cid:17) , (cid:16) ( ij ) (cid:16) j ( ij ) − i (cid:17) ( ij ) (cid:17) = (cid:16) ( ij )( ij ) − ( ij ) (cid:17) = ( ij ) , so (cid:16) j ( ij ) − i (cid:17) is an idempotent and (cid:16) j ( ij ) − i (cid:17) I ( ij ) . Using the fact that preinversesare unique so that idempotents are preinverses of themselves by Corollary 5.2, weconclude that ( ij ) is an idempotent since ( ij ) = (cid:16) j ( ij ) − i (cid:17) − = (cid:16) j ( ij ) − i (cid:17) .Finally, if ( ij ) is an idempotent then (( ij )( ij )) , so ( ji ) , and it can be shown inthe same way as for ( ij ) that ( ji ) is an idempotent. (cid:3) Proposition 6.5.
Let P be a pregroupoid. Then P has a unique preinverses if andonly if ( ij ) = ( ji ) for all idempotents i, j ∈ P such that ( ij ) .Proof. Let P have unique preinverses. If i, j ∈ P are idempotents and ( ij ) then ( ij ) and ( ji ) are idempotents by Lemma 6.1. Thus, (( ij )( ji )( ij )) = (( ij )( ij )) = ( ij ) , (( ji )( ij )( ji )) = (( ji )( ji )) = ( ji ) . so ( ij ) I ( ji ) . Hence, ( ji ) ∈ I {} ( ij ) = n ( ij ) − o = { ( ij ) } , using Corollary 5.2.Conversely, let idempotents in P commute. If y and y ′ are preinverses of x , then ( xyx ) = x and ( xy ′ x ) = x so that ( yx ) and ( y ′ x ) , so ( yx ) and ( y ′ x ) are idempotentssince ( yxyx ) = ( yx ) and ( y ′ xy ′ x ) = ( y ′ x ) . Similarly, ( xy ) and ( xy ′ ) are idempotents.We also have ( yxy ) = y and ( y ′ xy ′ ) = y ′ . Thus, y = ( yxy ) = ( yxy ′ xy ) = ( y ′ xyxy ) =( y ′ xy ) = ( y ′ xy ′ xy ) = ( y ′ xyxy ′ ) = ( y ′ xy ′ ) = y ′ . (cid:3) N GROUP-LIKE MAGMOIDS 13
Pregroupoids (2).
Proposition 6.6.
Let P be a pregroupoid P with unique preinverses, x ∈ P . Then λ x = λ − x and ρ x = ρ − x .Proof. As λ x and ρ x are idempotents by Proposition 6.2, we have λ x = λ − x and ρ x = ρ − x by Corollary 5.2. (cid:3) Proposition 6.7.
Let P be a pregroupoid with unique preinverses, x ∈ P . If x − is the preinverse of x then (cid:0) x − (cid:1) − = x .Proof. If x I x − then x − I x , so x ∈ I {} x − = n(cid:0) x − (cid:1) − o . (cid:3) Proposition 6.8.
Let P be a pregroupoid with unique preinverses, x, y ∈ P . If ( xy ) then ( xy ) − = (cid:0) y − x − (cid:1) .Proof. We have ( xy ) , (cid:0) xx − (cid:1) , (cid:0) x − x (cid:1) , (cid:0) yy − (cid:1) and (cid:0) y − y (cid:1) . Hence, by Proposition6.5 and the fact that (cid:0) x − x (cid:1) and (cid:0) yy − (cid:1) are idempotents, ( xy ) = (cid:0) xx − xyy − y (cid:1) = (cid:0) xyy − x − xy (cid:1) so that (cid:0) y − x − (cid:1) , and (cid:0) y − x − (cid:1) = (cid:0) y − yy − x − xx − (cid:1) = (cid:0) y − x − xyy − x − (cid:1) , so ( xy ) I (cid:0) y − x − (cid:1) , so (cid:0) y − x − (cid:1) ∈ I {} ( xy ) = n ( xy ) − o . (cid:3) If P is a pregroupoid with unique preinverses then (cid:0) x − (cid:1) − = x , so P can beequipped with a unique bijection i : P → P, x x − such that, for all x ∈ P , (( x i ( x )) x ) = x, ( x ( i ( x ) x )) = x, (6.3) (( i ( x ) x ) i ( x )) = i ( x ) , ( i ( x ) ( x i ( x ))) = i ( x ) . A pregroupoid with unique preinverses can thus be regarded as a prepoloid ( P, m , s , t ) expanded to a pregroupoid ( P, m , i , s , t ) characterized by the uniquenessproperty and the identities (4.1), (4.2), (6.1), and (6.3).If a pregroupoid with unique preinverses – or alternatively its reduct ( P, m , i ) –is a magma, it is thus an inverse semigroup, characterized by the uniqueness of thepreinverses and the identities(6.4) x ( yz ) = ( xy ) z, xx − x = x, x − xx − = x − . Note that a pregroupoid with unique preinverses does not necessarily have uniquelocal left and right units. While (cid:0) xx − (cid:1) and (cid:0) x − x (cid:1) are uniquely determined by x when its preinverse x − is unique, and are local left and local right units, respec-tively, for x , (cid:0) xx − (cid:1) and (cid:0) x − x (cid:1) are not necessarily the only local units for x . Example 6.1.
Let S be a set { x, y } with a binary operation given by the table x yx x xy x y. Let α, β, γ ∈ { x, y } . If α = β = γ = y then ( αβ ) γ = α ( βγ ) = y ; otherwise, ( αβ ) γ = α ( βγ ) = x . Thus, in all cases ( αβ ) γ = α ( βγ ) , so S is a semigroup. Inparticular, xxx = x and yyy = y , so x is a preinverse of x and y is a preinverseof y . Also, yxy = y , so x is not a preinverse of y , and y is not a preinverse of x .Hence, S is a pregroupoid where preinverses are unique. Local units are not unique, however; we have xx = x , so xx − = x is a local leftunit for x , but y is also a local left unit for x since yx = x . As a consequence, ifwe set s ( x ) = x, s ( y ) = y, s ( x ) = y, s ( y ) = y then ( s ( x ) x ) = ( s ( x ) x ) = x and ( s ( y ) y ) = ( s ( y ) y ) = y but s = s . Pregroupoids as prepoloids.
Let ( P, m , i ) be a reduct of the pregroupoid ( P, m , i , s , t ) .One can expand ( P, m , i ) to a pregroupoid ( P, m , i , s , t ) by setting s ( x ) = ( x i ( x )) and t ( x ) = ( i ( x ) x ) . Finally, we obtain the prepoloid ( P, m , s , t ) as a reduct of ( P, m , i , s , t ) ; below we note two useful results about this prepoloid in the case when s and t are unique. Recall from Section 4.4 that λ x = (cid:0) xx − (cid:1) and ρ x = (cid:0) x − x (cid:1) . Proposition 6.9.
Let P be a pregroupoid with unique canonical local units, x ∈ P .Then ( λ x λ x ) = λ x = λ λ x = ρ λ x and ( ρ x ρ x ) = ρ x = ρ ρ x = λ ρ x .Proof. Analogous to the proof of Proposition 6.2, in addition using the facts that (cid:0)(cid:0) xx − (cid:1)(cid:0) xx − (cid:1)(cid:1) = (cid:0) xx − (cid:1) and (cid:0)(cid:0) x − x (cid:1)(cid:0) x − x (cid:1)(cid:1) = (cid:0) x − x (cid:1) . (cid:3) Proposition 6.10.
Let P be a pregroupoid with unique canonical local units, x, y ∈ P . If ( xy ) then λ ( xy ) = λ x and ρ ( xy ) = ρ y .Proof. Analogous to the proof of Proposition 6.3. (cid:3)
The skew-prepoloid family.Definition 6.2.
Let P be a left (resp. right) semigroupoid. Then P is(1) a left (resp. right) skew-prepoloid when there is a local left (resp. right)unit λ x ∈ P (resp. ρ x ∈ P ) for every x ∈ P ; (2) a left (resp. right) skew-pregroupoid when P is a left (resp. right) skew-prepoloid such that for each x ∈ P there is a right (resp. left) preinverse x − of x in P .In view of the left-right duality of these notions, it suffices to consider left skew-prepoloids and left skew-pregroupoids here.By Definition 6.2, a left skew-prepoloid is a left semigroupoid P such that forevery x ∈ P there is some λ x ∈ P such that ( λ x x ) = x . A left skew-groupoid isa left skew-poloid P such that for every x ∈ P there is some x − ∈ P such that (cid:0) xx − (cid:1) ∈ { λ } x and (cid:0) x − x (cid:1) ∈ { λ } x − , so that (cid:0)(cid:0) xx − (cid:1) x (cid:1) = x and (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = x − . Left skew-prepoloids.
Proposition 6.11.
Let P be a left skew-prepoloid, x, y ∈ P . If λ y is a local leftunit for y and ( xλ y ) then ( xy ) .Proof. If ( xλ y ) then ( x ( λ y y )) = ( xy ) since ( λ y y ) = y . (cid:3) Proposition 6.12.
Let P be a left skew-prepoloid P with unique local units, x ∈ P .Then ( λ x λ x ) = λ x = λ λ x .Proof. We have x = ( λ x x ) = ( λ x ( λ x x )) = (( λ x λ x ) x ) , so ( λ x λ x ) ∈ { λ } x = { λ x } , so ( λ x λ x ) = λ x , so λ x ∈ { λ } λ x = { λ λ x } . (cid:3) Proposition 6.13.
Let P be a left skew-prepoloid P with unique local units, x, y ∈ P . If ( xy ) then λ ( xy ) = λ x .Proof. If ( xy ) then ( xy ) = (( λ x x ) y ) = ( λ x ( xy )) , so λ x ∈ { λ } ( xy ) = (cid:8) λ ( xy ) (cid:9) . (cid:3) By Definition 6.2, every left skew-prepoloid P with unique local left units canbe equipped with a unique surjective function s : P → { λ } P , x λ x N GROUP-LIKE MAGMOIDS 15 such that, for all x ∈ P ,(6.5) ( s ( x ) x ) = x. By Proposition (6.12), s ( λ x ) = λ x for all λ x ∈ { λ } P .A left skew-poloid with unique local units can thus be regarded as a left semi-groupoid ( P, m ) expanded to a left skew-poloid ( P, m , s ) characterized by the unique-ness property and the identities (4.1) and (6.5).We call a left semigroupoid P which admits a not necessarily unique function s : P → P satisfying (6.5) a left unital semigroupoid . If P is a magma then ( P, m , s ) is a semigroup such that, for all x, y, z ∈ P ,(6.6) x ( yz ) = ( xy ) z, s ( x ) x = x. Such a semigroup may be called a left unital semigroup . The class of left unitalsemigroups includes many types of semigroups studied in the literature, for exam-ple, D -semigroups [16], left abundant semigroups, left adequate semigroups, leftEhresmann semigroups, left ample semigroups and left restriction semigroups. Left skew-pregroupoids.
Proposition 6.14.
Let P be a left skew-pregroupoid with unique right preinverses, x ∈ P . Then λ − x = λ x .Proof. As λ x is an idempotent by Proposition 6.12, we have λ x ∈ I + {} x = (cid:8) λ − x (cid:9) as in Corollary 5.2. (cid:3) Proposition 6.15.
Let P be a left skew-pregroupoid with unique right preinverses, x ∈ P . If x − is the right preinverse of x then x = (cid:0) x − (cid:1) − .Proof. If x I + x − then x − I + x , so x ∈ I + {} x − = n(cid:0) x − (cid:1) − o . (cid:3) If P is a left skew-pregroupoid with unique right preinverses then (cid:0) x − (cid:1) − = x ,so P can be equipped with a unique bijection i : P → P, x x − such that, for all x ∈ P ,(6.7) (( x i ( x )) x ) = x, (( i ( x ) x ) i ( x )) = i ( x ) . A left skew-pregroupoid with unique right inverses can thus be regarded as a leftskew-poloid ( P, m , s ) expanded to a left skew-groupoid ( P, m , i , s ) characterized bythe uniqueness property and the identities (4.1), (6.5) and (6.7).Those left skew-pregroupoids with unique right preinverses which are magmasare again just inverse semigroups, characterized by the uniqueness of preinversesand the identities (6.4). Skew-pregroupoids as skew-prepoloids.
Let ( P, m , i ) be a reduct of ( P, m , i , s ) , andexpand ( P, m , i ) to a left skew-pregroupoid ( P, m , i , s ) by setting s ( x ) = ( x i ( x )) .We note two useful results about the left skew-prepoloid ( P, m , s ) , obtained as areduct of ( P, m , i , s ) , in the case when s is unique. Recall that λ x = (cid:0) xx − (cid:1) . Proposition 6.16.
Let P be a left skew-pregroupoid with unique canonical localleft units, x ∈ P . Then ( λ x λ x ) = λ x = λ λ x .Proof. Analogous to the proof of Proposition 6.12, although note that the presentproof requires the fact that (cid:0)(cid:0) xx − (cid:1)(cid:0) xx − (cid:1)(cid:1) = (cid:0)(cid:0)(cid:0) xx − (cid:1) x (cid:1) x − (cid:1) = (cid:0) xx − (cid:1) so that ( λ x λ x ) ∈ { λ } x . (cid:3) Proposition 6.17.
Let P be a left skew-pregroupoid with unique canonical localleft units, x, y ∈ P . If ( xy ) then λ x = λ ( xy ) .Proof. Analogous to the proof of Proposition 6.13. (cid:3) Poloids and related magmoids
The magmoids considered in this section differ from those in Section 6 in thatthey are equipped with two-sided or twisted units rather than local units. Thesemagmoids are, roughly speaking, categories and some of their specializations andgeneralizations, considered as algebraic structures. As noted in the introductionto this article, categories and groupoids are indeed examples of important notionsinvolving partial binary operations.7.1.
The poloid family.Definition 7.1.
Let P be a semigroupoid. Then P is(1) a poloid when there is a left effective unit ℓ x ∈ P and a right effective unit r x ∈ P for every x ∈ P ; (2) a groupoid when P is a poloid such that for every x ∈ P there is a strongpreinverse x − ∈ P of x .More explicitly, a poloid is a semigroupoid P such that for every x ∈ P thereare two-sided units ℓ x , r x such that ( ℓ x x ) = ( xr x ) = x , and also ( yℓ x ) = ( yr x ) =( ℓ x y ) = ( r x y ) = y for all x, y ∈ P such that, respectively, ( yℓ x ) , ( yr x ) , ( ℓ x y ) ,and ( r x y ) . A groupoid is a poloid P such that for every x ∈ P there is some x − ∈ P such that there are two-sided units ℓ x , r x such that (cid:0) xx − (cid:1) = ℓ x = r x − and (cid:0) x − x (cid:1) = r x = ℓ x − , so that (cid:0) xx − x (cid:1) = x , (cid:0) x − xx − (cid:1) = x − , andalso (cid:0) yxx − (cid:1) = (cid:0) yx − x (cid:1) = (cid:0) xx − y (cid:1) = (cid:0) x − xy (cid:1) = y for all x, y ∈ P such that,respectively, (cid:0) y (cid:0) xx − (cid:1)(cid:1) , (cid:0) y (cid:0) x − x (cid:1)(cid:1) , (cid:0)(cid:0) xx − (cid:1) y (cid:1) , and (cid:0)(cid:0) x − x (cid:1) y (cid:1) .It is shown below that every element of a poloid has unique left and right effectiveunits and a unique preinverse, and it turns out that poloids have much in commonwith prepoloids with unique local units and preinverses. A similar remark appliesto skew-poloids, defined below, in relation to skew-prepoloids.A poloid is just a (small) category regarded as an abstract algebraic structure[9], while a groupoid is a (small) category with preinverses, also regarded as anabstract algebraic structure. While categories are usually defined in another way,definitions similar to the definition of poloids given here can also be found in theliterature. For example, Ehresmann [2] proposed the following definition: Eine Kategorie ist eine Klasse C von Elementen, in der eine Multiplika-tion gegeben ist ( f, g ) → fg für gewisse Paare ( f, g ) von Elementen von C , welche folgenden Axiomen genügt:1. Wenn h ( fg ) oder ( hf ) g definiert ist, dann sind die beide Elementedefiniert und h ( fg ) = ( hf ) g .2. Wenn hf und fg definiert sind, dann ist auch h ( fg ) definiert.Ein Element e von C wird eine Einheit genannt, falls fe = f und eg = g für alle Elemente f und g von C ist, für welche fe und eg definiert sind.3. Für jedes f ∈ C gibt es zwei Einheiten α ( f ) und β ( f ) , so dass fα ( f ) und β ( f ) f definiert sind. (p. 50). Proposition 7.1 below implies that if a poloid is a semigroup then it is a monoid,since it has only one two-sided unit, denoted , and if a groupoid is a semigroupthen it is a group. Conversely, a poloid with just one two-sided unit is a monoid,and a groupoid with just one two-sided unit is a group [9]. A groupoid is thus ageneralized group, as expected, while poloids generalize groups indirectly and intwo ways, via monoids and via groupoids. N GROUP-LIKE MAGMOIDS 17
Poloids.
Proposition 7.1.
Let P be a poloid. If e, e ′ ∈ { e } P and ( ee ′ ) then e = e ′ .Proof. We have e = ( ee ′ ) = e ′ . (cid:3) Proposition 7.2.
Let P be a poloid, x ∈ P . Then there is a unique left effectiveunit ℓ x ∈ P and a unique right effective unit r x ∈ P for every x ∈ P .Proof. If ℓ x , ℓ ′ x ∈ { ℓ } x then x = ( ℓ x x ) = ( ℓ x ( ℓ ′ x x )) = (( ℓ x ℓ ′ x ) x ) , so ( ℓ x ℓ ′ x ) and thus ℓ x = ℓ ′ x . Dually, if r x , r ′ x ∈ { r } x then x = ( xr x ) = (( xr ′ x ) r x ) = ( x ( r ′ x r x )) , so ( r ′ x r x ) and thus r x = r ′ x . (cid:3) Proposition 7.3.
Let P be a poloid, x, y ∈ P . Then ( xy ) if and only if r x = ℓ y .Proof. If ( xy ) then ( xy ) = (( xr x )( ℓ y y )) = ((( xr x ) ℓ y ) y ) = (( x ( r x ℓ y )) y ) , so ( r x ℓ y ) ,so r x = ℓ y . Conversely, if r x = ℓ y then ( xℓ y ) , so ( x ( ℓ y y )) and thus ( x ( ℓ y y )) = ( xy ) since ( ℓ y y ) = y . (cid:3) Proposition 7.4.
Let P be a poloid. If e ∈ { e } P then ( ee ) = e = ℓ e = r e .Proof. We have e = ( ℓ e e ) = ℓ e and e = ( er e ) = r e . (cid:3) Corollary 7.1.
Let P be a poloid, x ∈ P . Then ( ℓ x ℓ x ) = ℓ x = ℓ ℓ x = r ℓ x and ( r x r x ) = r x = ℓ r x = r r x . Proposition 7.5.
Let P be a poloid, x, y ∈ P . If ( xy ) then ℓ ( xy ) = ℓ x and r ( xy ) = r y .Proof. We use the fact that effective units are unique by Proposition 7.2. If ( xy ) then ( xy ) = (( ℓ x x ) y ) = ( ℓ x ( xy )) , so ℓ x ∈ { ℓ } ( xy ) = (cid:8) ℓ ( xy ) (cid:9) . Dually, ( xy ) =( x ( yr y )) = (( xy ) r y ) , so r y ∈ { r } ( xy ) = (cid:8) r ( xy ) (cid:9) . (cid:3) In view of Definition (7.1) and Proposition 7.2, every poloid P can be equippedwith unique surjective functions s : P → { ℓ } P , x ℓ x , t : P → { r } P , x r x such that, for any x, y ∈ P , ( s ( x ) x ) = x, ( x t ( x )) = x ( s ( x ) y ) → ( s ( x ) y ) = y, ( t ( x ) y ) → ( t ( x ) y ) = y, (7.1) ( y s ( x )) → ( y s ( x )) = y, ( y t ( x )) → ( y t ( x )) = y. By Corollary 7.1, s ( ℓ x ) = ℓ x for all ℓ x ∈ { ℓ } P and t ( r x ) = r x for all r x ∈ { r } P .A poloid P can thus be regarded as an expansion ( P, m , s , t ) , characterized bythe identities (4.1), (4.2), and (7.1), of a semigroupoid ( P, m ) .If ( P, m , s , t ) is a magma then it degenerates to a monoid ( P, m , where s ( x ) = t ( x ) = 1 for all x ∈ P . Groupoids.
Proposition 7.6.
Let P be a poloid, x ∈ P . If x ( − is a strong pseudoinverse of x then x ( − is a strong preinverse of x .Proof. For any strong pseudoinverse x ( − of x , (cid:0) xx ( − (cid:1) ∈ { ℓ } x ⊆ { e } P and (cid:0) x ( − x (cid:1) ∈ { r } x ⊆ { e } P , so (cid:0) x ( − (cid:0) xx ( − (cid:1)(cid:1) = x ( − and (cid:0)(cid:0) x ( − x (cid:1) x ( − (cid:1) = x ( − .Thus, (cid:0) xx ( − (cid:1) ∈ { ρ } x − and (cid:0) x ( − x (cid:1) ∈ { λ } x − , so x ( − is a preinverse of x , andhence a strong preinverse of x . (cid:3) Hence, one may alternatively define a groupoid as a poloid P such that for every x ∈ P there is a strong pseudoinverse x ( − ∈ P of x , and definitions of this formare common in the literature. Proposition 7.7.
Let P be a poloid, x ∈ P . Then there is at most one strongpseudoinverse x ( − ∈ P of x .Proof. If x ′ and x ′′ are strong pseudoinverses of x then ( xx ′ ) ∈ { ℓ } x ⊆ { e } P and ( x ′′ x ) ∈ { r } x ⊆ { e } P , so x ′ = (( x ′′ x ) x ′ ) = ( x ′′ ( xx ′ )) = x ′′ . (cid:3) Corollary 7.2.
Let P be a groupoid. Then every x ∈ P has a unique strongpreinverse x − ∈ P . Proposition 7.8.
Let P be a groupoid. If e ∈ P is a two-sided unit then e − = e .Proof. The assertion is that { e } = I {} e . As ( ee ) = e we have (( ee ) e ) = ( e ( ee )) = e ,so e I e , so e ∈ I {} e = (cid:8) e − (cid:9) since preinverses are unique. Thus, { e } = I {} e , soto prove the assertion it suffices to note that (cid:0) ee − (cid:1) = (cid:0) e − e (cid:1) = ( ee ) = e , so that (cid:0) ee − (cid:1) and (cid:0) e − e (cid:1) are two-sided units. (cid:3) Proposition 7.9.
Let P be a groupoid, x ∈ P . Then (cid:0) x − (cid:1) − = x .Proof. If x I x − then x − I x , meaning that x is a strong preinverse of x − . Thus, x ∈ I {} x − = n(cid:0) x − (cid:1) − o since strong preinverses are unique. (cid:3) Proposition 7.10.
Let P be a groupoid, x, y ∈ P . If ( xy ) then ( xy ) − = (cid:0) y − x − (cid:1) .Proof. If ( xy ) then r x = ℓ x − = ℓ y = r y − , so (cid:0) y − x − (cid:1) . Furthermore, we have (cid:0) x − x (cid:1) ∈ { r x } ⊆ { e } P and (cid:0) yy − (cid:1) ∈ { ℓ y } ⊆ { e } P , so (cid:0) xyy − x − xy (cid:1) , and we obtain (cid:0) ( xy ) (cid:0) y − x − (cid:1) ( xy ) (cid:1) = (cid:0) x (cid:0) yy − (cid:1)(cid:0) x − x (cid:1) y (cid:1) = ( xy ) . Also, (cid:0) y − x − xyy − x − (cid:1) , and we have (cid:0)(cid:0) y − x − (cid:1) ( xy ) (cid:0) y − x − (cid:1)(cid:1) = (cid:0) y − (cid:0) x − x (cid:1)(cid:0) yy − (cid:1) x − (cid:1) = (cid:0) y − x − (cid:1) . Thus, ( xy ) I (cid:0) y − x − (cid:1) . Also, (cid:0) ( xy ) (cid:0) y − x − (cid:1)(cid:1) = (cid:0) x (cid:0) yy − (cid:1) x − (cid:1) = (cid:0) xx − (cid:1) ∈ { ℓ x } ⊆ { e } P , (cid:0)(cid:0) y − x − (cid:1) ( xy ) (cid:1) = (cid:0) y − (cid:0) x − x (cid:1) y (cid:1) = (cid:0) y − y (cid:1) ∈ { r y } ⊆ { e } P , so ( xy ) I (cid:0) y − x − (cid:1) . Hence, (cid:0) y − x − (cid:1) ∈ I {} ( xy ) = n ( xy ) − o since strong prein-verses are unique. (cid:3) Since every x ∈ P has a unique preinverse x − such that (cid:0) x − (cid:1) − = x , there isa unique bijection i : P → P, x x − such that, for any x, y ∈ P , (( x i ( x )) x ) = x, ( x ( i ( x ) x )) = x, (( i ( x ) x ) i ( x )) = i ( x ) , ( i ( x )( x i ( x ))) = i ( x ) , (7.2) (( x i ( x )) y ) → (( x i ( x )) y ) = y, ( y ( x i ( x ))) → ( y ( x i ( x ))) = y, (( i ( x ) x ) y ) → (( i ( x ) x ) y ) = y, ( y ( i ( x ) x )) → ( y ( i ( x ) x )) = y. A groupoid P can thus be regarded as an expansion ( P, m , i , s , t ) , characterizedby the identities (4.1), (4.2), (7.1), and (7.2), of a poloid ( P, m , s , t ) ,If ( P, m , i , s , t ) is a magma then it degenerates to a group ( P, m , i , where s ( x ) = t ( x ) = 1 for all x ∈ P . N GROUP-LIKE MAGMOIDS 19
The skew-poloid family.Definition 7.2.
Let P be a left (resp. right) semigroupoid. Then P is(1) a left (resp. right) skew-poloid when there is a unique twisted left (resp.right) unit ϕ x ∈ P (resp. ψ x ∈ P ) for every x ∈ P ; (2) a left (resp. right) skew-groupoid when P is a left (resp. right) skew-poloidsuch that for every x ∈ P there is a strong left (resp. right) preinverse x − ∈ P of x and a unique ϕ x − ∈ P (resp. ψ x − ∈ P ) such that if x − is a strong left (resp. right) preinverse of x then ϕ x − (resp. ψ x − ) is thetwisted left (resp. right) unit for x − .In view of the left-right duality in the skew-poloid family, it suffices to consideronly left skew-poloids and left skew-groupoids below. Note that we cannot in gen-eral regard (( xy ) z ) and ( x ( yz )) as equivalent expressions, written ( xyz ) , in thiscase; ( x ( yz )) implies (( xy ) z ) , but not conversely.By Definition 7.2, a left skew-poloid is a left semigroupoid P such that for every x ∈ P there is a unique ϕ x ∈ P such that ( ϕ x x ) = x and such that ( yϕ x ) = y for every y ∈ P such that ( yϕ x ) . A left skew-groupoid is a left skew-poloid P such that for every x ∈ P there is some x − ∈ P such that ϕ x = (cid:0) xx − (cid:1) and ϕ x − = (cid:0) x − x (cid:1) , so that (cid:0)(cid:0) xx − (cid:1) x (cid:1) = x , (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = x − , (cid:0) y (cid:0) xx − (cid:1)(cid:1) = y forall x, y ∈ P such that (cid:0) y (cid:0) xx − (cid:1)(cid:1) , and (cid:0) y (cid:0) x − x (cid:1)(cid:1) = y for all x, y ∈ P such that (cid:0) y (cid:0) x − x (cid:1)(cid:1) . In addition, because of the two uniqueness assumptions in Definition7.2, we have that if (( xx ′ ) x ) = x, (( x ′ x ) x ′ ) = x ′ and (( xx ′′ ) x ) = x, (( x ′′ x ) x ′′ ) = x ′′ then not only ( xx ′ ) = ( xx ′′ ) but also ( x ′ x ) = ( x ′′ x ) .A left (or one-sided) skew-poloid is what has been called a constellation [6,4, 5]. There is a close relationship between poloids and (left) skew-poloids, orbetween categories and constellations, because both notions formalize the idea ofa system of (structured) sets and many-to-one correspondences between these sets.Without going into details, the difference between the two notions is that in the firstcase many-to-one correspondences are formalized as functions , with domains andcodomains, whereas in the second case, many-to-one correspondences are formalizedas prefunctions , with domains but without codomains. Left skew-poloids.
Proposition 7.11.
Let P be a left skew-poloid. If ϕ, ϕ ′ ∈ { ϕ } P and ( ϕϕ ′ ) = ( ϕ ′ ϕ ) then ϕ = ϕ ′ .Proof. We have ϕ = ( ϕϕ ′ ) = ( ϕ ′ ϕ ) = ϕ ′ . (cid:3) Proposition 7.12.
Let P be a left skew-poloid, x, y ∈ P . Then ( xy ) if and only if ( xϕ y ) .Proof. If ( xy ) then ( xy ) = ( x ( ϕ y y )) = (( xϕ y ) y ) , and if ( xϕ y ) then ( x ( ϕ y y )) = ( xy ) since ( ϕ y y ) = y . (cid:3) Corollary 7.3.
Let P be a left skew-poloid, x, y ∈ P . Then ( xy ) if and only if ( xϕ y ) = x . Proposition 7.13.
Let P be a left skew-poloid, x ∈ P . Then ( ϕ x ϕ x ) = ϕ x = ϕ ϕ x .Proof. We have ϕ ϕ x = ( ϕ ϕ x ϕ x ) = ϕ x . (cid:3) Proposition 7.14.
Let P be a left skew-poloid, x, y ∈ P . If ( xy ) then ϕ ( xy ) = ϕ x .Proof. If ( xy ) then ( xy ) = (( ϕ x x ) y ) = ( ϕ x ( xy )) , so ϕ x ∈ { ϕ } ( xy ) = (cid:8) ϕ ( xy ) (cid:9) . (cid:3) For details, see [9, 5]. In [5], prefunctions are interpreted as surjective functions; the prefunc-tion f : X → Y is rendered as the function f : X → im f ⊆ Y . In view of Definition 7.2, every left skew-poloid P can be equipped with a uniquesurjective function s : P → { ϕ } P , x ϕ x such that, for any x, y ∈ P ,(7.3) ( s ( x ) x ) = x, ( y s ( x )) → ( y s ( x )) = y. By Proposition 7.13, s ( ϕ x ) = ϕ x for all ϕ x ∈ { ϕ } P .A left skew-poloid can thus be regarded as an expansion ( P, m , s ) , characterizedby the identities (4.1) and (7.3), of a left semigroupoid ( P, m ) .If ( P, m , s ) is a magma then it degenerates to a monoid ( P, m , where s ( x ) = t ( x ) = 1 for all x ∈ P . Left skew-groupoids.
Proposition 7.15.
Let P be a left skew-groupoid, x ∈ P . Then there is at mostone strong right preinverse x − ∈ P of x .Proof. Let x ′ and x ′′ be strong right preinverses of x . By the uniqueness of ϕ x , ϕ x ′ ,and ϕ x ′′ for all x ∈ P , we have ϕ x = ( xx ′ ) = ( xx ′′ ) as well as ϕ x ′ = ( x ′ x ) = ( x ′′ x ) = ϕ x ′′ . Thus, x ′ = (( x ′ x ) x ′ ) = (( x ′′ x ) x ′ ) = ( x ′′ ( xx ′ )) = ( x ′′ ( xx ′′ )) = (( x ′′ x ) x ′′ ) = x ′′ . (cid:3) Corollary 7.4.
Let P be a left skew-groupoid, x ∈ P . Then x has a unique strongright preinverse x − ∈ P . Proposition 7.16.
Let P be a left skew-groupoid. If ϕ ∈ { ϕ } P then ϕ − = ϕ .Proof. The assertion is that { ϕ } = I + {} ϕ . By Proposition 7.13, ( ϕϕ ) = ϕ , so (( ϕϕ ) ϕ ) = ϕ , so ϕ I + ϕ , so ϕ ∈ I + {} ϕ = (cid:8) ϕ − (cid:9) since strong right preinversesare unique. Thus, { ϕ } = I + {} ϕ , so to prove the assertion it suffices to note that (cid:0) ϕϕ − (cid:1) = (cid:0) ϕ − ϕ (cid:1) = ( ϕϕ ) = ϕ , so that (cid:0) ϕϕ − (cid:1) , (cid:0) ϕ − ϕ (cid:1) ∈ { ε } P . (cid:3) Proposition 7.17.
Let P be a left skew-groupoid, x ∈ P . Then (cid:0) x − (cid:1) − = x .Proof. If x I + x − then x − I + x , meaning that x is a strong right preinverse of x − .Thus, x ∈ I + {} x − = n(cid:0) x − (cid:1) − o since strong right preinverses are unique. (cid:3) Since every x ∈ P has a unique preinverse x − such that (cid:0) x − (cid:1) − = x , there isa unique bijection i : P → P, x x − such that, for all x, y ∈ P , (( x i ( x )) x ) = x, (( i ( x ) x ) i ( x )) = i ( x ) , (7.4) ( y ( x i ( x ))) → ( y ( x i ( x ))) = y, ( y ( i ( x ) x )) → ( y ( i ( x ) x )) = y. A left skew-groupoid can be thus be regarded as an expansion ( P, m , i , s ) , char-acterized by the identities (4.1), (7.3), and (7.4), of a left skew-poloid ( P, m , s ) .If ( P, m , i , s ) is a magma then it degenerates to a group ( P, m , i , where s ( x ) = 1 for all x ∈ P . If (( xy ) z ) then ( x ( yz )) since ( yz ) , so if (( xy ) z ) or ( x ( yz )) then (( xy ) z ) = ( x ( yz )) . N GROUP-LIKE MAGMOIDS 21 Prepoloids and pregroupoids with restricted binary operations
In a poloid, ( xy ) if and only if r x = ℓ y . In a prepoloid, ρ x = λ y implies ( xy ) byProposition 6.1, but ρ x = λ y is not a necessary condition for ( xy ) . If we retain onlythose products ( xy ) for which ρ x = λ y , we obtain a magmoid P [ m ] with the sameelements as P but restricted multiplication m : ( x, y ) ( x · y ) . By definition, wethen have ( x · y ) if and only if ρ x = λ y , as in a poloid, and under suitable conditionsthe restricted magmoid does indeed become a poloid.By similarly restricting the binary operation, we can derive a groupoid froma pregroupoid, a skew-poloid from a skew-prepoloid and a skew-groupoid from askew-pregroupoid.8.1. From prepoloids to poloids.Definition 8.1.
Let P be a prepoloid with binary operation m : ( x, y ) ( xy ) , x, y ∈ P . The restricted binary operation on the carrier set of P is a binaryoperation m : ( x, y ) ( x · y ) such that if ( x · y ) and ( xy ) then ( x · y ) = ( xy ) , and ( x · y ) if and only if relative to m there is some ρ x ∈ { ρ } x and some λ y ∈ { λ } y suchthat ρ x = λ y .In particular, Definition 8.1 applies to magmoids where λ x and ρ x are uniquelocal units for x ∈ P .Note that if ( x · y ) then ρ x = λ y , so ( xy ) , so if ( x · y ) then ( x · y ) = ( xy ) byDefinition 8.1. Lemma 8.1.
Let P be a prepoloid with unique local units. If λ x = λ λ x = ρ λ x and ρ x = λ ρ x = ρ ρ x for all x ∈ P and also λ ( xy ) = λ x and ρ ( xy ) = ρ y for all x, y ∈ P such that ( xy ) then P [ m ] is a poloid where ℓ x = λ x and r x = ρ x for all x ∈ P [ m ] .Proof. We first prove that P [ m ] is a semigroupoid. If ( x · ( y · z )) then ρ x = λ ( y · z ) and ρ y = λ z , so ρ x = λ ( y · z ) = λ ( yz ) = λ y . Thus, ρ ( x · y ) = ρ ( xy ) = ρ y = λ z , so (( x · y ) · z ) , so ( x · ( y · z )) = ( x ( yz )) = (( xy ) z ) = (( x · y ) · z ) .Dually, if (( x · y ) · z ) then ρ ( x · y ) = λ z and ρ x = λ y , so ρ y = ρ ( xy ) = ρ ( x · y ) = λ z .Thus, ρ x = λ y = λ ( yz ) = λ ( y · z ) , so ( x · ( y · z )) , so (( x · y ) · z ) = (( xy ) z ) = ( x ( yz )) =( x · ( y · z )) .Also, if ( x · y ) and ( y · z ) then ρ x = λ y and ρ y = λ z , so ρ x = λ y = λ ( yz ) = λ ( y · z ) and ρ ( x · y ) = ρ ( xy ) = ρ y = λ z . Hence, ( x · ( y · z )) and (( x · y ) · z ) , so ( x · ( y · z )) =( x ( yz )) = (( xy ) z ) = (( x · y ) · z ) .It remains to show that λ x and ρ x are two-sided units such that ( λ x · x ) = x and ( x · ρ x ) = x for all x ∈ P [ m ] . If ( λ x · y ) then λ x = ρ λ x = λ y , so ( λ x · y ) = ( λ x y ) =( λ y y ) = y , and if ( y · λ x ) then ρ y = λ λ x = λ x , so ( y · λ x ) = ( yλ x ) = ( yρ y ) = y .Similarly, if ( ρ x · y ) then ρ x = ρ ρ x = λ y , so ( ρ x · y ) = ( ρ x y ) = ( λ y y ) = y , and if ( y · ρ x ) then ρ y = λ ρ x = ρ x so ( y · ρ x ) = ( yρ x ) = ( yρ y ) = y .Thus, λ x and ρ x are two-sided units in P [ m ] for all x ∈ P [ m ] , and we have ( λ x · x ) = ( λ x x ) = x and ( x · ρ x ) = ( xρ x ) = x for all x ∈ P [ m ] . (cid:3) Combining Lemma 8.1 with Propositions 6.2 and 6.3 we obtain the followingresults:
Theorem 8.1. If P is a prepoloid with unique local units then P [ m ] is a poloid. Corollary 8.1. If P is a bi-unital semigroup with unique local units then P [ m ] isa poloid. From pregroupoids to groupoids.Definition 8.2.
Let P be a pregroupoid with binary operation m : ( x, y ) ( xy ) , x, y ∈ P . The restricted binary operation on the carrier set of P is a binaryoperation m : ( x, y ) ( x · y ) such that if ( x · y ) and ( xy ) then ( x · y ) = ( xy ) , and ( x · y ) if and only if there is a canonical local right unit ρ x = (cid:0) x − x (cid:1) for x and acanonical local left unit λ y = (cid:0) yy − (cid:1) for y such that ρ x = λ y .If ( x · y ) then there are ρ x , λ y such that ρ x = λ y , so ( xy ) by Proposition 6.1, so ( x · y ) = ( xy ) . Lemma 8.2.
Let P be a pregroupoid with unique canonical local units. If λ x = λ λ x = ρ λ x and ρ x = λ ρ x = ρ ρ x for all x ∈ P and also λ ( xy ) = λ x and ρ ( xy ) = ρ y for all x, y ∈ P such that ( xy ) then P [ m ] is a groupoid where ℓ x = λ x and r x = ρ x for all x ∈ P [ m ] .Proof. We can prove that P [ m ] is a poloid where ℓ x = λ x and r x = ρ x for all x ∈ P [ m ] by using the argument in the proof of Lemma 8.1 again. It remains toshow that every x ∈ P [ m ] has a strong preinverse in P [ m ] . Let x − be a preinverseof x in P . As ρ x = (cid:0) x − x (cid:1) = λ x − we have (cid:0) x · x − (cid:1) , and as ρ x − = (cid:0) xx − (cid:1) = λ x we have (cid:0) x − · x (cid:1) , so (cid:0)(cid:0) x · x − (cid:1) · x (cid:1) and (cid:0) x · (cid:0) x − · x (cid:1)(cid:1) . Thus, (cid:0)(cid:0) x · x − (cid:1) · x (cid:1) = (cid:0)(cid:0) xx − (cid:1) x (cid:1) = x and (cid:0) x · (cid:0) x − · x (cid:1)(cid:1) = (cid:0) x (cid:0) x − x (cid:1)(cid:1) = x , so (cid:0) x · x − (cid:1) = ℓ x ∈ { e } P and (cid:0) x − · x (cid:1) = r x ∈ { e } P by the uniqueness of effective units in P [ m ] . Thus x − is astrong pseudoinverse of x in P [ m ] and hence a strong preinverse of x in P [ m ] byProposition 7.6. (cid:3) Combining Lemma 8.2 with Propositions 6.9 and 6.10, we obtain the followingresults:
Theorem 8.2. If P is a pregroupoid with unique canonical local units then P [ m ] is a groupoid. Corollary 8.2. If S is a regular semigroup with unique canonical local units then S [ m ] is a groupoid. Theorem 8.3. If P is a pregroupoid with unique preinverses then P [ m ] is agroupoid.Proof. If preinverses are unique then (cid:0) xx − (cid:1) and (cid:0) x − x (cid:1) are uniquely determinedby x , so canonical local units are unique. (cid:3) Corollary 8.3. If S is an inverse semigroup then S [ m ] is a groupoid. It is clear that Corollaries 8.2 and 8.3 are related to the so-called Ehresmann-Schein-Nampooribad theorem in semigroup theory: groupoids correspond to inversesemigroups and, more generally, to regular semigroups whose canonical local unitsare unique.8.3.
From left skew-prepoloids to left skew-poloids.Definition 8.3.
Let P be a left skew-prepoloid with a binary operation m : ( x, y ) ( xy ) , x, y ∈ P . The restricted multiplication on the carrier set of P is a binary operation m : ( x, y ) ( x · y ) such that if ( x · y ) and ( xy ) then ( x · y ) = ( xy ) , and ( x · y ) if and only if there is some local left unit λ y for y suchthat ( xλ y ) = x .Note that if ( x · y ) then ( xy ) by Proposition 6.11, so then ( x · y ) = ( xy ) . Lemma 8.3.
Let P be a left skew-prepoloid with unique local left units. If ( λ x λ x ) = λ x = λ λ x for all x ∈ P and λ ( xy ) = λ x for all x, y ∈ P such that ( xy ) then P [ m ] isa left skew-poloid where ϕ x = λ x for all x ∈ P [ m ] . N GROUP-LIKE MAGMOIDS 23
Proof. If ( x · ( y · z )) then ( yλ z ) = y , so x = (cid:0) xλ ( y · z ) (cid:1) = (cid:0) xλ ( yz ) (cid:1) = ( xλ y ) , so ( x · y ) ,so ( x · y ) = ( xy ) . Thus, ( xy ) = ( x ( yλ z )) = (( xy ) λ z ) , so (( xy ) · z ) = (( x · y ) · z ) .Hence, if (( x · ( y · z ))) then ( x · ( y · z )) = ( x ( yz )) = (( xy ) z ) = (( x · y ) · z ) .If ( x · y ) and ( y · z ) then ( xλ y ) = x and ( yλ z ) = y . Hence, x = ( xλ y ) = (cid:0) xλ ( yz ) (cid:1) = (cid:0) xλ ( y · z ) (cid:1) , so ( x · ( y · z )) , so again ( x · ( y · z )) = ( x ( yz )) = (( xy ) z ) =(( x · y ) · z ) .If ( y · λ x ) then ( yλ λ x ) = y so ( y · λ x ) = ( yλ x ) = y since λ x = λ λ x . We also have ( λ x λ x ) = λ x , so ( λ x · x ) , so ( λ x · x ) = ( λ x x ) = x . Thus, λ x is a right unit anda local left unit for x , that is, a twisted left unit ϕ x for x . The uniqueness of ϕ x follows from the uniqueness of λ x . (cid:3) Combining Lemma 8.3 with Propositions 6.12 and 6.13, we obtain the followingresults:
Theorem 8.4. If P is a left skew-prepoloid with unique local left units then P [ m ] is a left skew-poloid. Corollary 8.4. If S is a left unital semigroup with unique local left units then S [ m ] is a left skew-poloid. The fact that a left skew-poloid can be constructed from a left skew-prepoloidis related to the fact that an inductive constellation can be constructed from a leftrestriction semigroup [6, 4].8.4.
From left skew-pregroupoids to left skew-groupoids.Definition 8.4.
Let P be a left skew-pregroupoid with binary operation m : ( x, y ) ( xy ) , x, y ∈ P . The restricted binary operation on the carrierset of P is a binary operation m : ( x, y ) ( x · y ) such that if ( x · y ) and ( xy ) then ( x · y ) = ( xy ) , and ( x · y ) if and only if there is a canonical local left unit λ y = (cid:0) yy − (cid:1) such that ( x λ y ) = x .If ( x · y ) then ( x λ y ) so ( xy ) by Proposition 6.11, so ( x · y ) = ( xy ) . Lemma 8.4.
Let P be a left skew-pregroupoid with unique canonical local left units.If ( λ x λ x ) = λ x = λ λ x for all x ∈ P and λ ( xy ) = λ x for all x, y ∈ P such that ( xy ) then P [ m ] is a left skew-groupoid where ϕ x = λ x and ϕ x − = λ x − for all x ∈ P [ m ] .Proof. It can be proved that P is a left skew-poloid with unique twisted left units ϕ x = λ x by using the argument in the proof of Lemma 8.3 again. To completethe proof, we first show that for every x ∈ P [ m ] there is a corresponding rightpreinverse x − ∈ P [ m ] .If x − ∈ P is a right preinverse of x then (cid:0) xx − (cid:1) = λ x and (cid:0) x − x (cid:1) = λ x − .Thus, ( x λ x − ) = (cid:0) x (cid:0) x − x (cid:1)(cid:1) = (cid:0)(cid:0) xx − (cid:1) x (cid:1) = ( λ x x ) = x, (cid:0) x − λ x (cid:1) = (cid:0) x − (cid:0) xx − (cid:1)(cid:1) = (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = (cid:0) λ x − x − (cid:1) = x − , so (cid:0) x · x − (cid:1) and (cid:0) x − · x (cid:1) since λ x − = λ ( x − ) by the uniqueness of canonical localleft units (Section 4.4). Hence, (cid:0)(cid:0) x · x − (cid:1) · x (cid:1) and (cid:0)(cid:0) x − · x (cid:1) · x − (cid:1) , so (cid:0)(cid:0) x · x − (cid:1) · x (cid:1) = (cid:0)(cid:0) xx − (cid:1) x (cid:1) = x, (cid:0)(cid:0) x − · x (cid:1) · x − (cid:1) = (cid:0)(cid:0) x − x (cid:1) x − (cid:1) = x − , so x − is a right preinverse of x in P [ m ] .It is shown in the proof of Lemma 8.3 that if ( y · λ x ) then ( y · λ x ) = y for any x, y ∈ P . It can be shown by a similar argument that if ( y · λ x ) then ( y · λ x ) = y ,and that if ( y · λ x − ) then ( y · λ x − ) = (cid:0) y · λ ( x − ) (cid:1) = y . Hence, x − is a strongright preinverse of x in P [ m ] with ϕ x = λ x and ϕ x − = λ x − = λ ( x − ) .The uniqueness of ϕ x − follows from the uniqueness of λ ( x − ) . (cid:3) Combining Lemma 8.4 with Propositions 6.16 and 6.17, we obtain the followingresults:
Theorem 8.5. If P is a left skew-pregroupoid with unique canonical local left units,then P [ m ] is a left skew-groupoid. Theorem 8.6. If P is a left skew-pregroupoid with unique preinverses then P [ m ] is a left skew-groupoid.Proof. If preinverses are unique then (cid:0) xx − (cid:1) is determined by x , so canonical localleft units are unique. (cid:3) Final remarks.
Sections 6 and 7 are structured around three main distinc-tions, exemplified by the distinctions between poloids and skew-poloids, poloidsand prepoloids, and poloids and groupoids. These distinctions generate eighttypes of poloid-like magmoids: prepoloids, pregroupoids, skew-prepoloids, skew-pregroupoids, poloids, groupoids, skew-poloids, and skew-groupoids.The most superficial of the three main distinctions is perhaps the one betweenpoloid-like and skew-poloid-like magmoids. The equivalence of these two notionswas briefly discussed in Section 7.2. The idea that the difference between themultimately reflects the way mappings are formalized – as functions with domainsand codomains, corresponding to the two-sided objects, or as prefunctions withonly domains, corresponding to the one-sided objects – is implicit in [9]. Theclose connection between the two notions does not mean, though, that the corre-sponding distinction is trivial or of little interest; showing the material equivalenceof superficially different notions is an important accomplishment in mathematics.It seems that much work remains to be done in this connection.Next, the distinction between poloid-like and prepoloid-like magmoids is also adistinction between two closely related notions. We have seen in Section 8 that(skew-)prepoloids with unique local units and (skew-)pregroupoids with uniquepreinverses can be transformed into corresponding (skew-)poloids and (skew-)group-oids. Conversely, it is shown in the literature that prepoloid-like magmoids can berecovered from corresponding poloid-like magmoids with additional structure. Therelationship between prepoloid-like and poloid-like magmoids, in particular semi-groups and poloids/groupoids, has been researched extensively, but not exhaus-tively, starting with Ehresmann.Finally, there is the distinction between different kinds of poloid-like magmoidsexemplified by that between monoids and groups. This is obviously a significantdistinction, though one which hardly needs further comment here.
Appendix A. Heap-like algebras with partial operations
As we know, a group, with a binary operation ( x, y ) xy , is closely relatedto a corresponding heap with a ternary operation, written ( x, y, z ) [ x, y, z ] or ( x, y, z ) [ xyz ] . Analogously, an involution magmoid, with a partial binary op-eration ( x, y ) xy and total unary operation x x ∗ , is closely related to a cor-responding algebra with a partial ternary operation ( x, y, z ) [ xyz ] as describedbelow. Semiheapoids and semiheaps from involution semigroupoids.
One can use any totalinvolution ∗ on a semigroupoid P to define a partial ternary operation t : P × P × P → P, ( x, y, z ) [ xyz ] := ( xy ∗ z ) on P , with [ xyz ] being defined if and only if ( xy ∗ z ) . In particular, a transformation magmoid is associative, whereas a pretransformation mag-moid is just skew-associative; see Facts 3 and 4 in [9].
N GROUP-LIKE MAGMOIDS 25
Let [[ xyz ] uv ] , [ xy [ zuv ]] and [ x [ uzy ] v ] be defined. Then [[ xyz ] uv ] = (( xy ∗ z ) u ∗ v ) =( xy ∗ zu ∗ v ) , [ xy [ zuv ]] = ( xy ∗ ( zu ∗ v )) = ( xy ∗ zu ∗ v ) and [ x [ uzy ] v ] = (cid:0) x ( uz ∗ y ) ∗ v (cid:1) = (cid:0) x (cid:0) y ∗ ( z ∗ ) ∗ u ∗ (cid:1) v (cid:1) = ( xy ∗ zu ∗ v ) . Thus,(A.1) [[ xyz ] uv ] = [ x [ uzy ] v ] = [ xy [ zuv ]] . Extending Wagner’s terminology [18], we call any set P with a partial ternary oper-ation t : ( x, y, z ) [ xyz ] satisfying (A.1) when all terms are defined a semiheapoid .If t is a total function satisfying (A.1) then P is a semiheap . Semiheapoids and semiheaps from groupoids; heapoids and heaps.
By Propositions7.9 and 7.10, every groupoid has a total involution, namely the function x x − .In this case, the identities (7.2) give rise to identities applying to t . Specifically, if (cid:0) xx − y (cid:1) then (cid:0) xx − y (cid:1) = ( ℓ x y ) = y and if (cid:0) yx − x (cid:1) then (cid:0) yx − x (cid:1) = ( yr x ) = y , so(A.2) [ xxy ] = [ yxx ] = y. We call a non-empty set P with a partial ternary operation t satisfying (A.1)and (A.2) when all terms are defined a heapoid . If t is a total function satisfying(A.1) and (A.2) then P is a heap . Semiheapoids and semiheaps from pregroupoids; preheapoids and preheaps.
By Pro-positions 6.7 and 6.8, every pregroupoid P with unique preinverses has a totalinvolution, namely the function x x − . If the involution has this form then (cid:0) yy − zz − x (cid:1) = (cid:0) zz − yy − x (cid:1) , (cid:0) xy − yz − z (cid:1) = (cid:0) xz − zy − y (cid:1) since idempotents in pregroupoids with unique preinverses commute, so [ yy [ zzx ]] = [ zz [ yyx ]] , [[ xyy ] zz ] = [[ xzz ] yy ] . (A.3)We also have (cid:0) xx − x (cid:1) = x , so(A.4) [ xxx ] = x. We call a non-empty set P with a total function t satisfying (A.1), (A.3) and (A.4)a preheap or, in Wagner’s terminology, a generalized heap; if t is a partial functionsatisfying (A.1), (A.3) and (A.4) when all terms are defined, we get a preheapoid instead. Generalized semiheapoids.
We can let not only the binary operation on a semi-groupoid P but also the involution ∗ on P be a partial function. Then [ xyz ] isdefined if and only if ( x ( y ∗ ) z ) , and then [ xyz ] = ( x ( y ∗ ) z ) . Such generalized semi-heapoids can be defined naturally for semigroupoids of matrices (see Section 5.2). References [1] Burmeister K (1992). Partial algebras – survey of a unifying approach towards a two-valuedmodel theory for partial algebras.
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Mathematische Annalen , Communicationsin Algebra , Doklady AkademiiNauk SSSR, Mate-maticheskii sbornik, Dan Jonsson, Department of Sociology and Work Science, University of Gothen-burg, SE 405 30 Gothenburg, Sweden.
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