aa r X i v : . [ m a t h . R A ] J a n Skew Meadows
November 19, 2018J A Bergstra Informatics Institute, University of Amsterdam,Science Park 403, 1098 SJ Amsterdam, The NetherlandsY Hirshfeld Department of Mathematics, Tel Aviv University,Tel Aviv 69978, IsraelJ V Tucker Department of Computer Science, Swansea University,Singleton Park, Swansea, SA2 8PP, United Kingdom
Abstract
A skew meadow is a non-commutative ring with an inverse operator satisfying twospecial equations and in which 0 − = 0. All skew fields and products of skew fieldscan be viewed as skew meadows. Conversely, we give an embedding of non-trivialskew meadows into products of skew fields, from which a completeness result forthe equational logic of skew fields is derived. The relationship between regularityconditions on rings and skew meadows is investigated. Keywords . Field, skew field, meadow, skew meadow, embedding theorem, ini-tial algebras, equational specifications, regular ring, strongly regular ring, inversesemigroup.
Meadows have been introduced in [8] while the equations for meadows used below wereimproved in [3, 4]. Meadows may be viewed as a generalization of so-called zero totalizedfields, being fields in which division is made total by setting 0 − = 0. Thanks to acharacterization theorem in [3], however, a meadow can also be defined as a commutativering with unit equipped with a total unary operation x − , named inverse, that satisfiesthese two additional equations: ( x − ) − = x (1) x · ( x · x − ) = x (2) Email: [email protected] Email: [email protected] Email: [email protected]
Ref , the second equation is called
Ril .Perhaps the clearest way to specify the class of meadows is as the smallest varietycontaining all zero-totalized fields. This is a matter of taste to some extent.Commutative meadows provide an analysis of division which is more general thanthat of the classical theory of fields. Commutative meadows are total algebras in which0 − = 0. We have used algebras with such zero totalized division in developing elementaryalgebraic specifications for several algebras of rational numbers in our previous paper [8]and its companions [2, 9].Several generalizations of meadows can be conceived. In [5] signed meadows weredefined as a generalization of ordered fields (with totalized division), and in [6] differentialmeadows are considered which may be viewed as a generalization of differential fields.The generalization of commutative meadows, as defined in [8] and subsequently ana-lyzed in [4], to the non-commutative case is the subject of this paper. As is always thecase, the transition from commutative to non-commuutative rings is a delicate operation,leading to a ramification of properties. However, we are able isolate a number of conceptsand prove nice generalizations of basic results, including the following RepresentationTheorem 4.13: Theorem
An algebra is a non-trivial skew meadow if, and only if, it is isomorphic toa subalgebra of a product of zero totalized skew fields.
Ring theory has several concepts, like Von Neumann regularity, that distinguish elementswith properties similar to multiplicative inverses, but does not seem to have investigatedthe possible corresponding inverse operators. We establish the relationship between skewmeadows and forms of von Neumann regularity conditions on non-commutative rings(Theorem 4.14). The equational nature of meadows is confirmed for skew meadows bythis result (Theorem 4.15):
Theorem
An equation is valid in all skew fields with zero totalized division if, and onlyif, it is true in the variety of skew meadows and for that reason derivable from their equa-tional axiomatisation.
Investigations of fields with zero totalized division have some history in logic and comput-ing: it is mentioned in [12] as a reasonable method to extend division to a total function,and it has been used in a more technical way in [11]. This work may be viewed as belong-ing to universal algebra and equational logic, with some orientation towards computerscience, in particular, to the theory of abstract data types. We refer to [17] and [20]for computer science oriented introductions to universal algebra. Of course the paper isabout noncommutative rings. We refer to [18] and [15] for introductions to noncommuta-tive rings. To some extent the results are intimately related to the theory of regular andinverse semigroups, because many of the arguments can be given without a reference toaddition and subtraction [16]. We will focus, however, on connections with the theory ofnoncommutative rings. 2
Axioms for rings
We start with a listing of the axioms of a unital ring. The starting point is a signatureΣ RU for rings with unit: signature Σ RU sorts ring operations → ring ;1 : → ring ;+ : ring × ring → ring ; − : ring → ring ; · : ring × ring → ring end The first set of axioms is that of a ring with 1, which establishes the standard prop-erties of +, − , and · . equations RU ( x + y ) + z = x + ( y + z ) (3) x + y = y + x (4) x + 0 = x (5) x + ( − x ) = 0 (6)( x · y ) · z = x · ( y · z ) (7)1 · x = x (8) x · ( y + z ) = x · y + x · z (9)( x + y ) · z = x · z + y · z (10) end These axioms generate a wealth of properties of + , − , · with which we will assume thereader is familiar. We will write x − y as an abbreviation of x + ( − y ). We notice that x · x is not implied by these axioms, whereas x · · x = 0 are derivable:0 = x · − x · x · (0 + 0) − x · x · x · − x · x · x · − x ·
0) = x · A ring is commutative if it satisfies : ∀ x ∃ y. ( x · y = y · x ) . A ring is called von Neumann regular (regular for short) if it satisfies: ∀ x ∃ y. ( x · y · x = x ) .
3n element y with x · y · x = x is called a pseudoinverse of x . Moreover, an element y with y · x = 1 is called an inverse of x . Indeed every inverse is a pseudoinverse as well.Following e.g., [14]: a ring is strongly regular if it satisfies: ∀ x ∃ y. ( x · x · y = x ) . A regular ring is called unit regular if it satisfies: ∀ x ∃ y ∃ z. ( x · y · x = x & y · z = 1) . An idempotent is an element e of a ring that satisfies e · e = e . An element is c is central if it satisfies ∀ x. ( x · c = c · x ).There is an equivalent definition of strong regularity that is closer to our objectives:a ring is strongly regular if it is regular and its idempotents are central.An element x of a ring is nilpotent if some power of it equals 0, i.e., ∃ n.x n = 0 . A ring is reduced if it has no non-zero nilpotent elements, i.e., ∀ x. ( x · x = 0 = ⇒ x = 0) . It is immediate that strongly regular rings are reduced. One can prove as a corollary ofProposition 4.5 below, that a reduced regular ring is strongly regular.In [1] it is shown (using the first definition) that a strongly regular ring is regular.According to [19] in a strongly regular ring every idempotent is central. Therefore bothdefinitions coincide. Below we will need this information, but we will provide completeproofs and we will indicate how in an alternative and shorter exposition use could havebeen made from this equivalence.We notice that the central elements of a ring constitute a subring. Further, in com-mutative rings regularity and strong regularity coincide. A skew field , also called a division ring , is a unital ring that satisifies the general inverselaw ( Gil ): ∀ x = 0 . ∃ y. ( y · x = 1) . A division ring is a division algebra if it is finitely generated over its centralizer subring.The quaternions as designed by Hamilton constitute a division algebra. A commutativeskew field is called a field .In skew fields, 1 is a right unit as well as a left unit, and left inverses are also rightinverses. Every skew field is a strongly regular ring. Indeed consider x , then if x = 0, wehave x = x · x · y (for any y ) and if x = 0 then there is some y with x · y = 1 for whichof course x = x · x · y . Skew fields are also unit regular rings. Consider again x : if x = 0then x = x · · · x = 0 and y its inverse: x · y · x = x · y · x = 1. 4 .3 The intended meaning of meadows and skew meadows Meadows and skew meadows are concepts created for the following purpose: meadows aresupposed to be models of the equational theory of zero totalized fields and skew meadowsare supposed to be models of the equational theory of zero totalized skew fields (i.e.,skew fields with zero totalized division). Thus, they have an intended meaning that isindependent of their axiomatic definition.It is due to Theorem 4.5, that the intended meaning of skew meadows is capturedby the technical definition that we will provided below. The corresponding fact wasestablished for meadows in [3].The intended meaning of meadows and skew meadows implies that if one considersstructures which violate equations about division true in division algebras these algebraswill not be called meadows. A typical example of such an equation is x · x − = x − · x . Analgebra that fails to comply with this equation cannot be a meadow or a skew meadow. To the signature Σ RU of rings with unit, we add an inverse operator − to form the pri-mary signature Σ Md , which we will use for totalized division rings and meadows: signature Σ Md import Σ RU operations − : ring → ring end As we insist that all operations are interpreted as total functions x − must be definedfor all x . A division operator which has been made total is called a totalized divisionoperator. Fields and skew fields can be enriched to Σ Md algebras by extending the signa-ture and defining 0 − = a for some appropriate a . A (skew) field thus obtained is called a -totalized . In the sequel we will only consider the case a = 0 and work with zero totalizedfields and skew fields.A skew meadow is a Σ RU algebra which satisfies RU and in addition these two equa-tions: equations SkMd import RU , Σ Md ( x − ) − = x (11) x · ( x · x − ) = x (12) end Axiom 11 is called
Ref for reflection . Axiom 12 is called
Ril for restricted inverse law .Together these axioms imply 0 − = 0 because 0 − = 0 − · − · (0 − ) − = 0 − · − · meadow is a commutative skew meadow. In terms of axioms, the equations for skewmeadows result from the equations for meadows, by simply dropping commutativity ofmultiplication and including a second distributivity law; this is the same modificationthat is needed to move from unital commutative rings to arbitrary rings with a left unit.A zero totalized field is a meadow and a zero totalized skew field is a skew meadow.Just as meadows capture the equational theory of fields with zero totalized division, ina similar fashion, skew meadows have been designed to capture the equational theory ofskew fields with zero totalized division.A skew meadow is (an enrichment of) a strongly regular ring. A meadow is an enrich-ment of a commutative regular ring (which must be strongly regular as well). In [3] itwas shown that every commutative regular ring can be expanded with an inverse operatorto a meadow. Below, in Proposition 4.14, we will find that in the noncommutative caseevery strongly regular ring can be expanded to a skew meadow.The non-commutative case makes distinctions between axioms that are useful equiv-alents in the commutative case. For example, the equation ∀ x. ( x · x − · x = x )is called Pil and cannot be assumed to be equivalent with
Ril . Forgetting 0, + and - a meadow restricts to an inverse semigroup, which is a specialcase of regular semigroups. Thus a meadow combines a commutative ring and an inversesemigroup and a skew meadow combines a possibly noncommutative ring with an inversesemigroup. The theory of inverse semigroups provides information about the existenceof an inverse operation in regular semigroups. From that perspective the information inour Propositions 4.9 and 4.14 below that strongly regular rings can be uniquely equippedwith an inverse operator that satisfies both laws is known as a result in semigroup theory.We have included the results and proofs for completeness of our exposition.Inverse semigroups have been introduced to formalize properties of function spaces.More specifically, inverse semigroups are to partial symmetries what groups are to sym-metries [16]. We have not found evidence that inverse semigroups have been used toformalize totalized division in the mathematical literature.
In this section we provide a number of logical and structural results concerning skewmeadows. We start with some preparations.
When working with meadows we will frequently use some auxiliary notation in order toimprove readability or conciseness. We formalize the use of these inessential notations byincluding them in an extended specification
Aux . Of course, these operators can always6e removed from any specification or proof in favor of their explicit definitions. signature Σ Aux sorts ring operations − : ring → ring ; − : ring → ring ;1 − : ring × ring → ring ; Z : ring → ring ; −− : ring × ring → ring endequations Aux import Σ Md , Σ Aux x = x · x (13) x = x − (14)1 x = x · x (15) Z ( x ) = 1 − x (16) xy = x · y (17) end In this section we will state and prove a number of useful facts valid in the variety of skewmeadows.
Proposition 4.1. x · x = 0 → x = 0 .Proof. If x · x = 0 then also x · x · x − = 0 · x − = 0. Now using Ril one obtains x = 0. Proposition 4.2. x − · x − · x = x − .Proof. Combine
Ril and
Ref . Proposition 4.3. x · y = 1 → x = y − .Proof. Assume x · y = 1, then 1 = x · y = x · y · y · y − = 1 · y · y − = y · y − . Therefore x = x · x · ( y · y − ) = ( x · y ) · y − = 1 · y − = y − .An obvious and useful consequence of this proposition is 1 − = 1. Proposition 4.4. x · x − is an idempotent: ( x · x − )( x · x − ) = x · x − . roof. We have: x · x − = ( x · x · x − ) · x − and therefore ( x · x − ) = ( x · x · x − · x − ) = x · x · ( x − · x − · x ) · x · x − · x − = ( x · x · x − ) · x · x − · x − = ( x · x · x − ) · x − = x · x − . Proposition 4.5. (i) ∀ x. ( x · x ) ;(ii) ∀ x. ( x · x − · x = x ) (i.e., Pil ); and,(iii) every idempotent is central: if e · e = e then ∀ x ( e · x = x · e ) .Proof. This proof uses the following idea taken from [10] (chapter 3). For every y andevery idempotent e we have: [ e · y · (1 − e )] = 0 as (1 − e ) · e = 0, and for the same reasonalso [(1 − e ) · y · e ] = 0. Therefore we have e · y · (1 − e ) = 0 and (1 − e ) · y · e = 0.Using distributivity we find e · y · − e · y · e = 0 and 1 · y · e − e · y · e = 0. Combiningthese two equations we have: e · y · · y · e and then e · y · y · e .In particular with e = y · y − we have y · y − · y · y · y · y − , and using Ril we find y · y − · y · y . Right-multiplying both sides with y · y − we have y · y − · y · · y · y − = y · y · y − and then y · y − · y · (1 · y · y − ) = y . Now using RU we have y · y − · y · ( y · y − ) = y which can be written as y · y − · ( y · y · y − ) = y . Now Ril gives y · y − · y = y (i.e., Pil ).Combining this fact with the equation y · y − · y · y · y · y − that was established aboveone finds y · y for arbitrary y . Now using the equation e · y · y · e which has alreadybeen established, we have e · y = y · e as desired.This proof makes use only of the fact that the meadow is reduced. An alternative(and shorter) proof using the literature cited in Section 2.1 is as follows: the meadow isa strongly regular ring which must therefore be regular (which proves Pil ) and moreoverits idempotents are central. Because 1 is an idempotent it is central and therefore x · · x = x . Proposition 4.6. x · x − = x − · x .Proof. x · x − = x · ( x − · x − · x ) = ( x · x − · x − ) · x = [( x · x − ) · x − ] · x =[ x − · ( x · x − )] · x = x − · x . Corollary 4.7.
Every skew meadow is a Dedekind finite ring, i.e., it satisfies: x · y = 1 → y · x = 1 .Proof. If x · y = 1 then y = x − and thus y · x = x − · x = x · x − = x · y = 1. Proposition 4.8.
Skew meadows are expansions of unit regular rings.Proof.
Given x , if 1 x = 1 then x − is a unit. Otherwise, let y = (1 − x ) + x − and y ′ = (1 − x )+ x . Now x · y · x = x · · x − x · x · x + x · x − · x = x · x − x · x + x = x and moreover: y · y ′ = [(1 − x )+ x − ] · [(1 − x )+ x ] = (1 − x ) · (1 − x )+(1 − x ) · x + x − · (1 − x )+ x − · x =1 − x − x +1 x · x + x − x · x + x − − x − · x + x − · x = 1 − x − x +1 x + x − x + x − − x − +1 x =1 . Proposition 4.9.
The inverse function is uniquely determined in a strong sense: For agiven x if there is some y such that: ( x · y · x = x or x · x · y = x ) and ( y · x · y = y or y · y · x = y ), then y = x − . roof. If x · y · x = x then x · y is idempotent so that x · x · y = x also holds. The sameholds for y · x · y = y . Therefore we have in any case that x · x · y = x and yyx = y .We denote e = x · x − and make use the fact that it is an idempotent and that itcommutes with every element, that it is equal to x − x , and that x = ex : y = y · y · x = y · y · ( e · e · e · x ) = e · e · y · e · y · x = x − · x − · x · x · y · e · y · x = x − · x − · x · e · y · x = x − · x − · x − · x · x · y · x = x − · x − · x − · x · x = x − · x − · x = x − . Proposition 4.10. (i) x · x = x = 1 x · x (ii) x · y · x · y = 1 x · y (iii) x · y = x · y · y · x (iv) x · y = y · x .Proof. (i) is just Ril , (and for the second equality
Pil ).(ii)1 x · y · x · y = x · y · x · y · x · y = x · x · y · y · x · y = x · y · x · y = 1 x · y . (iii) immediate.(iv) x · y = x · y · x · y · x · y = x · y · x · x · x · y · y = x · y · x · y · x · x · y = x · y · x · y · y · x · x · y = x · y · y · x · x · y = x · y · y · x · x · x · y · y = x · y · x · y · y · x · x · y = x · y · x · y · y · x · y · x · x · y = x · y · x · y · y · x · y · x · y = x · y · x · y · x · y · x · y · y = x · y · ( x · y ) · x · y · x · y = x · y · ( x · y ) · x · y · y · x = x · y · ( x · y ) · y · x · y · x = x · y · y · x · y · x = 1 x · y · y · x = 1 x · y · x = y · x · x = y · x . Proposition 4.11. x · y = 1 y · x .Proof. x · y = x · y · x · y = x · y · y · x = x · y · x = x · x · y = 1 x · y = 1 y · x = 1 y · x . Corollary 4.12.
Every skew meadow satisfies: x · y = 0 → y · x = 0 .Proof. If x · y = 0 then y · x = y · x · y · x = y · x · x · y = y · x · x · y · x · y = y · x · · x · y = 0. Because the axioms for meadows are equations, and as they hold in all skew fields, allsubalgebras of products of skew fields are skew meadows.
Generalizing the results obtainedin [3] to the noncommutative case, we show the converse:
Theorem 4.13.
Every non-trivial skew meadow is isomorphic to a Σ Md substructure ofa product of zero totalized skew fields.Proof. The plan of the proof is as follows. Let R be a skew meadow in which 0 and 1are different. For every element x = 0 there is a homomorphism (with respect to allthe operations, including the inverse) of the structure R into a zero totalized skew field R x , which takes x away from 0. These homomorphisms can then be combined into amonomorphism of R into the product of the skew fields R x fo all non-zero x .The proof is done in three steps: given an element x ∈ R we do the following:1. Define a ring homomorphism h from R onto the ring e x · R with its new unit 1 x ,and show that x is mapped to an invertible element.2. Prove that a skew meadow modulo a maximal ideal is a ring with division. If p isthe projection of the skew meadow 1 x · R onto this ring with division then p · h is a ringhomomorphism onto a skew field that maps x away from 0.9. We conclude the proof by showing that a ring homomorphism from the reductof a skew meadow into a skew field is a meadow homomorphism when the inverse isappropriately defined on the skew field.From now on an ideal in a ring will mean a two sided (left and right) ideal. Step 1 : Because 1 x is an idempotent in the meadow R it is central, and for that reason1 x · R is a (two sided) ideal, 1 x is a unit in this ideal, x and x − are in this ideal, inverseto each other, and h ( z ) = 1 x · z is a ring homomorphism. In 1 x · R the interpretation ofinverse as the mapping of 1 x · z to e x · z − makes the ring 1 x · R into a meadow. The proofis easy, using the axioms of a skew meadow and the properties in the propositions above. Step 2 : Let S be a skew meadow and J a maximal (two sided) ideal. Then S/J is adivision ring, and (trivially) every invertible element of S has a non zero image. J is a maximal ideal. Assume that x is not in J and look at the subset J + 1 x · S ,which is also a (two sided ideal). Therefore it is all of S , and for some r ∈ S and j ∈ J wehave j + 1 x · r = 1. Since 1 x = x · x − (and also x − · x ) we conclude that in the quotient[ x ] · [ x − · r ] = [1]. Therefore every non zero element in S/J has an inverse.
Step 3 : If H is a ring homomorphism from a skew meadow into a skew field then H preserves also inverses. Therefore it is a meadow homomorphism.If H ( x ) = 0 then H (1 x ) = H ( x · x − ) = H ( x ) · H ( x − ) = 0 so that also H ( x − ) = H (1 x · x − ) = H (1 x ) · H ( x − ) = 0 = ( H ( x )) − . We assume therefore that H ( x ) = 0.Then H ( x ) = H (1 x · x ) = H (1 x ) · H ( x ) which proves that H (1 x ) = 1, by cancellationin skew fields. In other words 1 = H ( x · x − ) = H ( x ) · H ( x − ), which proves that H ( x − ) = ( H ( x )) − . By Proposition 4.5 every skew meadow is an expansion of a strongly regular ring. Ournext observation is the converse: In every strongly regular ring there is a unique way tochoose for each element one of its inverse elements and define an inverse function whichis also reflexive:
Theorem 4.14. If R is a strongly regular ring then there is an inverse function (neces-sarily unique by Proposition 4.9) that turns it into a skew meadow.Proof. We will show first that for all the elements y that are pseudo inverse to x , i.e., xyx = x the idempotent xy is the same element and the element xyx is the same element.We wil then show that with the definition x − = yxy we obtain a skew meadow. Weproceed in four steps.a) If y and y ′ are pseudoinverses of x then x · y = x · y ′ . Indeed, first notice that x · y is idempotent: ( x · y ) · ( x · y ) = ( x · y · x ) · y = x · y . Then we calculate as follows: x · y ′ = ( x · y · x ) · y ′ = ( x · y ) · ( x · y ′ ) = ( x · y ′ ) · ( x · y ) = ( x · y ′ · x ) · y = x · y .If y is a pseudo inverse of x then we call the product x · y (which is independent ofthe choice of y ) the local unit of x , and we denote it by 1 lx . Trivially 1 lx · x = x · lx andeasily 1 lx · lx = 1 lx . 10) For every pseudo inverse y of x , 1 lx · y is also a pseudo inverse of x . Indeed: x · (1 lx · y ) · x = ( x · lx ) · y · x = x · y · x = x . Moreover, for every pseudo inverse y the product 1 lx · y yields the same pseudo inverse element. First notice that for a pseudoinverse y of y · x is an idempotent: ( y · x ) · ( y · x ) = y · ( x · y · x ) = y · x . Suppose that u , y and y ′ are pseudo inverses of x then 1 lx · y = ( x · u ) · y = y · ( x · u ) = y · ( x · y ′ · x ) · u =( y · x ) · ( y ′ · x ) · u = ( y ′ · x ) · ( y · x ) · u = y ′ · ( x · y · x ) · u = y ′ · ( x · u ) = ( x · u ) · y ′ = 1 x · y .Let y be any pseudo inverse for x we define x − as follows: x − = 1 lx · y . It has justbeen demonstrated that this definition is independent from the choice of y . With y apseudo inverse for x we find 1 x = x · x − = x · lx · y = x · ( x · y ) · y = ( x · y ) · x · y = x · y = 1 lx .c) By (b) x − is a pseudo inverse of x , and for all pseudo inverses y we have 1 x · y = x − therefore: 1 x · x − = x − . Moreover we know that x − · ( x − ) − is and idempotent. Weuse this to show that 1 x = 1 x − as follows: 1 x − = x − · ( x − ) − = 1 x · x − · ( x − ) − = x · x − · ( x − · ( x − ) − ) = x · ( x − · ( x − ) − ) · x − = x · ( x − · ( x − ) − · x − ) = x · x − = 1 x .d) After these preparations we can show that this inverse operation turns the ringinto a skew meadow. Ril is trivial since x − is a pseudo inverse of x , and it remains toshow that Ref also holds and therefore ( x − ) − = x . Indeed: ( x − ) − = ( x − ) − · x − =( x − ) − · x = ( x − ) − · x − · x = 1 x − · x = 1 x · x = x . As in [3], a completeness result follows from the embedding theorem.
Theorem 4.15.
Every equation valid in all skew fields with zero totalized division is truein the variety of skew meadows and for that reason derivable from SkMd.Proof.
A trivial skew meadow satisfies all equations and for that reason all equations validin all skew fields. Now consider a non-trivial skew meadow. According to Theorem 4.13it is embedded in a subalgebra of a product of skew fields. Now equations true of all skewfields are true in products of skew fields and in all subalgebras of such products includingthe given non-trivial skew meadow.Another interesting consequence of the embedding theorem is that a non-trivial skewmeadow must be infinite . According to Wedderburn’s Small Theorem, non-commutativeskew fields must have characteristic 0. A product of algebras with characteristic 0 hascharacteristic 0 as well, which implies that its minimal subalgebra is infinite.
Inversion rings are like skew meadows but without the restriction that idempotents arecentral. Because idempotents are central in all skew fields, inversion rings with non-centralidempotents will not be referred to as meadows. The ring that underlies an inversion ringmust be regular but need not be strongly regular.Because strong regularity is implied by the axioms for skew meadows some weakeningof the axioms needs to take place. Instead of
Ril we will use a (potentially) weaker axiom11hich we have called
Pil for pseudo inverse law . In Proposition 4.5 it was noticed thatskew meadows satisfy
Pil . The connection between the various notions is now as follows:skew meadows are inversion rings with central idempotents and meadows are commutativeskew meadows.The phrase inversion ring can be easily adapted to fields as follows: an inversion field is a zero totalized field and a skew inversion field is a zero totalized skew field. We noticethat an inversion skew field is an inversion ring in which all idempotents are central andwhere the only idempotents are 0 and 1.A generalization of an inversion skew field arises as follows: a semi-inversion skewfield is an inversion skew field in which the only central idempotents are 0 and 1. Thisdefinition gives rise to the question:
Can the embedding theorem and the completenesstheorem be generalized to semi-inversion skew fields . Stated in other words: Is the equa-tional theory of semi-inversion skew fields finitely based?
These axioms are weaker than the axioms for skew meadows. equations IR import RU , Σ Md x · x (18)( − x ) − = − ( x − ) (19)( x − ) − = x (20) x · ( x − · x ) = x (21) end An inversion ring is a model of IR . It is immediate that x · x − is an idempotent inan inversion ring. It is also the case that 0 − = 0 in any inversion ring. We have included( − x ) − = − ( x − ) because this equation expresses a very important symmetry. It is easilyderivable from the axioms of skew meadows. Currently we have no proof that it is notderivable from the other axioms of inversion rings. The same holds for x · x . An inversion ring is pseudo-commutative if it satisfies: ∀ x ∀ y. [( x · y ) − = y − · x − ] . Af-ter removing a redundant axiom one obtains the following axiomatization of pseudo-commutative inversion rings. equations
PCIR , Σ Md mport RU ( x · y ) − = y − · x − (22)( x − ) − = x (23) x · ( x − · x ) = x (24) end A ring will be called distinctly regular if its satisfies the following property: ∀ x ∃ ! y. ( x · y · x = x & y · x · y = y ) . It is immediate that x · x = x → x = x − in any distinctly regular ring. Strongly regularrings are distinctly regular as a corollary to Proposition 4.14. The importance of distinctregularity for inversion rings is implied by the next Proposition. Proposition 5.1.
A distinctly regular ring can be expanded to an inversion ring.Proof.
Obviously on defines x − as the unique y such that x · y · x = x and y · x · y = y . Ril is immediately satisfied. We will prove the other two axioms. Consider u = x − · x · u · x = x · x − · (1 · x ) = x · x − · x = x and u · x · u = x − · (1 · x ) · x − · x − · x · x − · x − · u . Thus u = x − for all x . Now substituting x − for x we find x = ( x − ) − = u − = u − · x − ) − · x · u = ( − x ) − . We have ( − x ) · u · ( − x ) = − x and u · ( − x ) · u = u fromthis we find x · ( − u ) · x = x and − u · x · − u = − u . Distinct regularity implies − u = x − which implies u = − x − which is the required fact.We notice that it follows from distinct regularity that the expansion is unique. Proposition 5.2.
A regular ring in which all idempotents commute is distinctly regular.Proof.
The result and its proof are valid in semigroups but we phrase both in terms ofrings. let R be a regular ring in which idempotents commute. Then every element ofS has at least one inverse. Suppose that a in S has two pseudoinverses b and c , i.e., a · b · a = a, b · a · b = b, a · c · a = a and c · a · c = c . Then a · b, a · c. b · a and c · a areidempotents as immediate consequences of these assumptions.Now b = b · a · b = b · ( a · c · a ) · b = b · a · c · ( a · c ) · ( a · b ) = b · a · c · ( a · b ) · ( a · c ) =( c · a ) · ( b · a ) · b · a · c = c · a · b · a · c = c · a · c = c. Proposition 5.3.
In a distinctly regular all idempotents commute.Proof.
This result is known in semigroup theory (see [13] Thm. 5.1.1) but we include aproof to make the paper self contained.We assume that for every element x in the ring there is a unique y such that x · y · x = y and y · x · y = x . This y is called the inverse of x and is denoted by x − . We notice that:1. If y = x − then x = y − , since the same pair of equations testify to both.2. If e is an idempotent then e − = e , since e = e · e · e is the only condition involved.13 laim 1: Let e be an idempotent. If e · x = x then x − · e = x − . If x · e = x then e · x − = x − . Proof: x · ( x − · e ) · x = x · x − · ( e · x ) = x · x − · x = x . On the other hand x − · e · x · x − · e = [ x − · ( e · x ) · x − ] · e = [ x − · x · x − ] · e = x − · e . Therefore x − · e isinverse to x and by uniqueness it equals x − . The other direction is similar. Claim 2: If e and f are idempotents then so is also e · f . Proof:
Let I = ( e · f ) − . We have e · e · f = e · f so that I · e = I , by lemma 1.Similarly f · I = I . Therefore I = I · e · f · I = ( I · e ) · ( f · I ) = I · I . Hence I is idempotentso that I = I − = e · f and in particular e · f is idempotent.Now we can prove that any two idempotents commute. Let e and f be idempotents.Then e · f is an idempotent and ( e · f ) − = e · f . We show that · e is also inverse to e · f sothat e · f = f · e by the uniqueness of inverse. Indeed ( e · f ) · ( f · e ) · ( e · f ) = e · ( · f ) · ( e · e ) · f = e · f · e · f = e · f , as e f and e · f are idempotents. Similarly ( f · e ) · ( e · f ) · ( f · e ) = f · e . Proposition 5.4.
A regular ring with commuting idempotents can be expanded to apseudo-commutative inversion ring.Proof.
Consider elements x and y . We have x · y · ( y − · x − ) · x · y = x · ( y · y − ) · ( x − · x ) · y = x · ( x − · x ) · ( y · y − ) · y = x · y and similarly ( y − · x − ) · x · y · ( y − · x − ) = y − · x − . In view ofProposition 5.2 the ring is distinctly regular and we can infer that ( x · y ) − = y − · x − . Proposition 5.5.
In a pseudo-commutative inversion ring where all idempotents e satisfy e = e − products of idempotents are idempotent.Proof. Consider idempotents e and f . We have e · f = e · f · ( e · f ) − · e · f = e · f · f − · e − · e · f = e · f · f · e · e · f = e · f · e · f . Proposition 5.6.
A distinctly regular inversion ring is pseudo-commutative.Proof.
Using Proposition 5.3 the ring has commuting idempotents. Now using 5.4 it mustbe pseudo-commutative as the expansion with an inverse operator must be unique.Several questions remain unanswered:
Is there a finite equational specification of theclass of directly regular inversion rings? Is there a finite equational specification of theclass of regular inversion rings in which all idempotents are equal to their own inverse?
We call a ring inversion compatible if it can be expanded with an inverse operator intoan inversion ring. Proposition 5.1 implies that distinctly regular rings are inversion com-patible. Obviously every inversion ring is an expansion of a regular ring. Trivially everyinversion compatible ring can be expanded to an inversion ring. We have not been ableto answer the following question:
Are all regular rings inversion compatible?
We expectthis not to be the case. 14e will now examine a particular example in meticulous detail. It proves the existenceof a non strongly regular and non-pseudo-commutative inversion ring. It can be concludedthat the replacement of Ril by Pil constitutes a weakening of the axioms .We have not yet developed any structure theory for non-central inversion rings. Alltechnical work on skew meadows depends on centrality and non-central inversion ringshave to be investigated from scratch.
The 2 × ( Q ) constitute a non-commutativering. Although familiar to all mathematicians we will spell out the details of this matterbecause of the importance of this example. A matrix X has the form X = (cid:18) x x x x (cid:19) . Let, in addition to X , the matrix Y be given by: Y = (cid:18) y y y y (cid:19) . Then we have the two constants for unital rings:0 = (cid:18) (cid:19) , (cid:18) (cid:19) and operators − X = (cid:18) − x − x − x − x (cid:19) , X + Y = (cid:18) x + y x + y x + y x + y (cid:19) and X · Y = (cid:18) x · y + x · y x · y + x · y x · y + x · y x · y + x · y (cid:19) An important observation is that (cid:18) (cid:19) · (cid:18) (cid:19) = (cid:18) (cid:19) . Clearly x · x = 0 → x = 0 fails in matrix rings. Because Ril implies x · x = 0 → x = 0,irrespectively of how an inverse is defined no expansion of the matrix ring to a skewmeadow is possible. There might however be an expansion possible to a very skew meadow.It will now be established that this is indeed the case. The matrix X is called regular if Det ( X ) = x · x − x · x = 0. We abbreviate D = Det ( X ) and if X is regular it is invertible and an explicit formula for the inverseexists: X − = (cid:18) x D − x D − x D x D (cid:19) .
15n this case we have: ( X − ) − = X , X · X − = X − · X = 1 and for that reason also Pil : X · ( X − · X ) = X .We will now define a suitable inverse in all other cases as well. Nonregular matricesmay have either four zeros (the case X = 0), three zeros (which splits in four cases), twozeros (either in the same column or in the same row), or no zeros at all and the secondcolumn equal to a scalar product of the first column. Together these are seven cases whichcan be dealt with (almost) independently. In the case of four zeros we take 0 − = 0 whichclearly satisfies both Ref and
Pil . In the case of three zeros we have x = 0 and the twodiagonal cases: (cid:18) x
00 0 (cid:19) − = (cid:18) x −
00 0 (cid:19) , (cid:18) x (cid:19) − = (cid:18) x − (cid:19) In these cases
Ril and
Pil are immediate and moreover X · X − = X − · X . Then considerboth remaining non-diagonal three-zero cases: (cid:18) x (cid:19) − = (cid:18) x − (cid:19) , (cid:18) x (cid:19) − = (cid:18) x − (cid:19) Now notice: (cid:18) x (cid:19) · (cid:18) x − (cid:19) = (cid:18) (cid:19) , and (cid:18) x − (cid:19) · (cid:18) x (cid:19) = (cid:18) (cid:19) At this stage it is apparent that the we are not dealing with an Abelian skew meadowbecause (cid:18) (cid:19) · (cid:18) (cid:19) = (cid:18) (cid:19) · (cid:18) (cid:19) In both non-diagonal cases
Ref is immediate and in the first case
Pil follows from: (cid:18) x (cid:19) · (cid:18) (cid:19) = (cid:18) x (cid:19) = (cid:18) (cid:19) · (cid:18) x (cid:19) . In the second non-diagonal case a similar calculation works for
Pil : (cid:18)(cid:18) x (cid:19) · (cid:18) x − (cid:19)(cid:19) · (cid:18) x (cid:19) = (cid:18) (cid:19) · (cid:18) x (cid:19) = (cid:18) x (cid:19) . There are four cases with two zeros, writing 2 for 1+1, which are pairwise inverses. Wehave x = 0 and y = 0. (cid:18) x y (cid:19) − = (cid:18) (2 · x ) − · y ) − (cid:19) , (cid:18) x y (cid:19) − = (cid:18) (2 · x ) − (2 · y ) − (cid:19) . Again
Ref is immediate. For
Pil we check: (cid:18)(cid:18) x y (cid:19) · (cid:18) (2 · x ) − · y ) − (cid:19)(cid:19) · (cid:18) x y (cid:19) = (cid:18) (cid:19) · (cid:18) x y (cid:19) = (cid:18) x y (cid:19) , (cid:18)(cid:18) x y (cid:19) · (cid:18) · x · y (cid:19)(cid:19) · (cid:18) x y (cid:19) = (cid:18) x · yy · x (cid:19) · (cid:18) x y (cid:19) = (cid:18) x y (cid:19) . The other two cases with two zeros are these: (cid:18) x y (cid:19) − = (cid:18) · x ) − · y ) − (cid:19) , (cid:18) x y (cid:19) − = (cid:18) · x ) − (2 · y ) − (cid:19) . Again
Ref is immediate. For
Pil we check: (cid:18)(cid:18) x y (cid:19) · (cid:18) · x ) − · y ) − (cid:19)(cid:19) · (cid:18) x y (cid:19) = (cid:18) (cid:19) · (cid:18) yx y (cid:19) = (cid:18) x y (cid:19) , and (cid:18)(cid:18) x y (cid:19) · (cid:18) · x · y (cid:19)(cid:19) · (cid:18) x y (cid:19) = (cid:18) x · yy · x (cid:19) · (cid:18) x y (cid:19) = (cid:18) x y (cid:19) . Finally the case of a nonregular matrix with all elements non-zero works as follows: (cid:18) x x · yx · z x · y · z (cid:19) − = (cid:18) · x · x · z · x · y · x · y · z (cid:19) Ref follows by means of elementary calculation. For
Pil we check: (cid:18)(cid:18) x x · yx · z x · y · z (cid:19) · (cid:18) · x · x · z · x · y · x · y · z (cid:19)(cid:19) · (cid:18) x x · yx · z x · y · z (cid:19) = (cid:18)
12 12 · zz (cid:19) · (cid:18) x x · yx · z x · y · z (cid:19) = (cid:18) x x · yx · z x · y · z (cid:19) . The example constitutes an inversion ring which is not pseudo-commutative. Indeedconsider the element P = (cid:18) (cid:19) .P is an idempotent but its inverse is not. In a pseudo-commutative inversion ring inversesof idempotents must be idempotent as well: e − = ( e · e ) − = e − · e − .We have yet to find an example of a pseudo-commutative inversion ring which is nota skew meadow. As we have stated already in Section 3.1 a skew meadow can be restricted by forgettingaddition and subtraction, thus obtaining a semigroup. Regular semigroups are defined assemigroups which satisfy: ∀ x ∃ y. ( x · y · x = x ). Clearly every inversion ring is an enrichmentof a regular semigroup. 17 more resticted class of semigroups consists of the inverse semigroups. An inversesemigroup is a semigroup that satisfies: ∀ x ∃ ! y. ( x · y · x = x & y · x · y = y )Here ∃ ! y.φ asserts the existence of a unique y such that φ . Clearly each inverse semigroupis regular. From Proposition 4.14 we find that strongly regular rings are enrichments (by0, + and -) of inverse semigroups. Thus any skew meadow is a combination of a reducedring and an inverse semigroup (and conversely).If a ring is an enrichment of an inverse semigroup it is obviously inverse compatible.We don’t know whether or not the converse is true.Of course inversion compatibility can be viewed as a property of semigroups ratherthan as a property of rings. All inverse semigroups are inversion compatible, but theconverse fails. Now a regular ring is inversion compatible precisely if its multiplicativesemigroup is inversion compatible.Inversion compatibility for regular semigroups is probably a more easily accessibletopic than inversion compatibility for regular rings. In this section thee algebraic specifications will be proposed each extending the specifica-tion of skew meadows.
The rational number specification from [8] can be weakened and commutativity is notessential because 1 x is central in any skew meadow, and that restricted form of commuta-tivity satisfies to prove the initiality result. We refer to [7, 8] for the definitions of initialalgebra specifications and initial algebra semantics. equations Zero totalized rationals import
SkMd, Aux Z (1 + x + y + z + u ) = 0 (25) end We will not repeat the argument of [8] proving that an initial algebra specification ofzero totalized rationals is obtained. Some comments are in order, however. The mainthrust of the proof is to demonstrate that each closed term t is provably equal to aterm in a set of canonical forms called the transversal. The transversal consists of 0,and k · ( l ) − , − k · ( l ) − for k and l relatively prime positive natural numbers and with n denoting the numeral for n , i.e., the sum of n t . We will only consider the cases t = r · s and t = r + s and ignore negationssigns to simplify notation. Suppose that r = k · ( l ) − and s = m · ( n ) − . As n and m are18ums of four squares plus 1 it can be shown that l · ( l ) − = n · ( n ) − = 1. Further it iseasy to prove that for all m and n , m · n = m.n .Now r + s = k · ( l ) − + m · ( n ) − = k · · ( l ) − + m · · ( n ) − = k · n · ( n ) − · ( l ) − + m · l · ( l ) − · ( n ) − = k · n · ( l · n ) − + m · l · ( n · l ) − = k.n · ( l.n ) − + m.l · ( n.l ) − = k.n + m.l · ( n.l ) − = p ′ .p · ( p ′′ .p ) − = p ′ · p · ( p ) − · ( p ′′ ) − = p ′ · · ( p ′′ ) − = p ′ · ( p ′′ ) − . Here k.n + m.l = p ′ .p and n.l = p ′′ .p with p the GCD of k.n + m.l and l.n .Next we will consider t = r · s . With induction on n one derives that for all n : n · x = x · n (i.e., all numerals are central in the initial algebra). Now r · s = k · ( l ) − · m · ( n ) − = k · m · ( l ) − · ( n ) − , and for terms of this form the previous part of the proof has alreadyestablished a path towards the canonical form.One may wonder whether a simpler specification is possible for instance by usingthe equation Z (1 + x ) = 0 instead. Now this will not work because the prime field ofcharacteristic 3 satisfies Z (1+ x ) = 0 but fails to be a homomorphic image of the meadowof rational numbers which is immediate by considering the image of 1 = (1 + 1 + 1) / (1 +1 + 1) which must equal 0 and 1 simultaneously. (At the time of writing we do not knowwhether or the equation Z (1 + x + y ) = 0 suffices to specify the rational numbers inthis context). As an exercise we specify the zero totalized complex rationals. This specification is anadaptation of the specification presented in [8]. c () is the complex conjugate. It servesas a unary auxiliary function. Just like the rationals the zero totalized complex rationalsconstitute a commutative meadow while the required amount of commutativity followsfrom the axioms for skew meadows already. The adaptation of the proof follows the samelines as in the case of the rational numbers. equations Zero totalized complex rationals import
SkMd, Aux operationsi : → ring ; c : ring → ring ; i = − c ( i ) = − i (27) c ( x − ) = c ( x ) − (28) c ( x + y ) = c ( x ) + c ( y ) (29) c ( x · y ) = c ( x ) · c ( y ) (30)1 c ( x ) = 1 x (31) Z (1 + x · c ( x ) + y · c ( y )) = 0 (32) end c () as an auxiliary operator. This specification suggestsan obvious question which was first mentioned in [9]: can a specification of the zerototalized complex rational numbers be given that makes no use of any auxiliary function. The quaternions are a well-known skew field. The rational quaternions constitute its primesub(skew)field. Its expansion to a skew meadow constitutes the zero totalized quaternions.This is a computable algebra. According to [7] every computable data type has an initialalgebra specification which may make use of auxiliary functions. Here we will make useof a unary auxiliary function c . As it turns out many algebras can be specified by meansof an initial algebra specification using a only single unary auxiliary function. We are notaware of any theoretical results that indicate why this is the case and for what kind ofalgebras a the use single unary auxiliary function will not be sufficient for giving an initialalgebra specification.The auxiliary function c ( − ) is the conjugate for quaternions. It is a division ring(pseudo) homomorphism which sends i , j , k to - i , - j and - k respectively.The following set of equations, together with the axioms on inverse and the equationsfor rings, specifiy the skew field of rational quaternions with zero totalized division as itsinitial algebra. Interestingly the sum of four squares in the equation that asserts the exis-tence of inverses is now implicit in the multiplication of a quaternion with its conjugate.We omit the correctness proof because it follows the general pattern as given in [9, 8] forthe rationals with out any significant complications. equations Zero totalized rational quaternions import
SkMd, Aux operationsi : → ring ; j : → ring ; k : → ring ; c : ring → ring ; i = − j = − k = − i · j · k = − c ( i ) = − i (37) c ( j ) = − j (38) c ( x − ) = c ( x ) − (39) c ( x + y ) = c ( x ) + c ( y ) (40) c ( x · y ) = c ( y ) · c ( x ) (41)1 c ( x ) = 1 x (42) Z (1 + x · c ( x )) = 0 (43)20 nd From these equations one easily proves: c (0) = 0, c (1) = 1, c ( k ) = − k j = − i · j = k , j · k = i , k · i = j , j · i = − k , k · j = − i , i · k = − j , and c ( − x ) = − c ( x ) andmany other well-known facts about quaternions. Just like in the case of complex numbersthe operation c may be viewed as an auxiliary operator in spite of the fact that it is avery familiar one. In both cases (complex rationals and rational quaternions) the questionwhether or not a specification can be given without an auxiliary operator is open. The generalization of our results on meadows in [3, 4] to the noncommutative cases isquite satisfactory. Many issues are left open, however, notably the development of astructure theory for non-central inversion rings. Another line of further work is to specifynonassociative algebras with a zero totalized division operator: zero totalized octonions.Nonassociative multiplication is relevant for the subject of division by zero also if onecontemplates alternatives containing some form of infinity value that will serve as a properinverse of zero.It is easy to see that the first order theory of fields is 1-1 reducible to the first ordertheory of meadows which in its turn is 1-1 reducible to the first order theory of skewmeadows. Because the first order theory of fields is undecidable so is the first ordertheory of meadows and so is the first order theory of skew meadows.For the equational theory of meadows at least these questions are currently open:(i) can one successfully perform a Knuth-Bendix completion, and: (ii) is the equationaltheory of meadows decidable. The same questions can be posed for skew meadows. For theaxioms of pseudo-commutative inversion rings, as well as the axioms for inversion rings,once more the same questions can be posed, such as what rings are inversion compatible,i.e., can be expanded to an inversion ring?
References [1]
R F Arens and I Kaplansky , Topological representations of algebras,
Trans. Amer.Math. Soc.
63 (1948), pp. 457-481.[2]
J A Bergstra , Elementary algebraic specifications of the rational function field, In
Logicalapproaches to computational barriers, Proceedings of CIE 2006
A. Beckman et. al. (eds.)Springer Lecture Notes in Computer Science vol. 3988, Springer-Verlag, New York, pp.40-54 (2006).[3]
J A Bergstra, Y Hirshfeld and J V Tucker , Fields, meadows and abstract datatypes,
Arnon Avron, Nachum Dershowitz and Alexander Rabinowitz (eds.), Pillars of Com-puter Science, (Essays dedicated to Boris Thaktenbroth on the occasion of his 85’th birth-day)
Springer Lecture Notes in Computer Science vol. 4800, Springer-Verlag, New York,pp. 166-178 (2008). J A Bergstra, Y Hirshfeld and J V Tucker , Meadows and the equational specifi-cation of division,
Theoretical Computer Science (2008), DOI: 10.1016/j.tcs.2008.12.015[5]
J A Bergstra and A Ponse
J A Bergstra and A Ponse
J A Bergstra and J V Tucker , Equational specifications, complete term rewritingsystems, and computable and semicomputable algebras,
J. ACM , 42 (1995) 1194-1230.[8]
J A Bergstra and J V Tucker , The rational numbers as an abstract data type,
J.ACM , 54, 2, Article 7 (April 2007) 25 pages.[9]
J A Bergstra and J V Tucker , Elementary algebraic specifications of the rationalcomplex numbers, In
Algebra, meaning and computation. Goguen Festschrift , K Futatsugiet. al. Eds. Lecture Notes in Computer Science, vol. 4060, Springer-Verlag, New York (2006)459-475.[10]
K R Goodearl , Von Neumann regular rings , Pitman, London, 1979.[11]
J Harrison , Theorem proving with the real numbers , Springer-Verlag New York, 1998.[12]
W Hodges , Model Theory , Cambridge University Press, Cambridge, 1993.[13]
J M Howie , Fundamentals of Semigroup Theory , Clarendon Press, Oxford, 1995.[14]
C Jayaram , Strongly regular rings,
Acta Mathematica Hungarica
56 (3-4), 1990, pp. 255-257.[15]
T Y Lam , A first course in the theory of noncommutative rings , Springer graduate textsin mathematics, 2001.[16]
M V Lawson , Inverse semigroups: The theory of partial symmetries , World Scientific,1998[17]
K Meinke and J V Tucker , Universal algebra, in S. Abramsky, D. Gabbay and TMaibaum (eds.)
Handbook of Logic in Computer Science. Volume I: Mathematical Struc-tures , Oxford University Press, 1992, pp.189-411.[18]
N McCoy , The theory of rings , Macmillan, London, 1964.[19]
M Satyanaryana , A note on p.p. rings,
Math Scand.
25, 1969, pp. 105-108.[20]
W Wechler , Universal algebra for computer scientists , EATCS Monographs in ComputerScience, Springer, 1992., EATCS Monographs in ComputerScience, Springer, 1992.