Associativity of the ∇-operation on bands in rings

Karin Cvetko-Vah and Jonathan Leech

Abstract

Given a multiplicative band of idempotents S in a ring R, for alle, f ∈ S the ∇-product e∇f = e+f +fe−efe−fef is an idempotentthat lies roughly above e and f in R just as ef and fe lie roughlybelow e and f . In this paper we study ∇-bands in rings, that is, bandsin rings that are closed under ∇, giving various criteria for ∇ to beassociative, thus making the band a skew lattice. We also considerwhen a given band S in R generates a ∇-band.

Given a set S of commuting idempotents in a ring R, it is well-knownthat under both multiplication x·y and the circle operation x◦y = x+y−xy,S will generate a set S that forms a lattice of idempotents in R with themeet e ∧ f given by ef and the join e ∨ f given by e ◦ f . (As is custom-ary, we will suppress the use of ‘·’ except when speaking of the operation.)What occurs if we begin with a noncommuting set S of idempotents in R?In general, S need not be closed under multiplication nor even generate alarger set of idempotents S that is thus closed. Indeed, it need not even bethe case that S2 = {ef | e, f ∈ S} ⊆ E(R), the set of all idempotents in R.Having admitted this, one is free to consider those sets S of idempotentssuch that S2 ⊆ E(R) at least or perhaps even generate a multiplicativelyclosed subset of E(R). Some of these cases might even admit a join-likeidempotent-producing operation j(x, y) with outputs lying above both in-puts in a manner similar to the case of lattices of commuting idempotentsin rings.

A semigroup consisting entirely of idempotents is called a band. A mul-tiplicative semigroup of idempotents in any ring R will be called “a band inR” in this paper. Bands in rings have been a focus of attention over the lasttwo decades. One group of researchers has studied bands in matrix rings,their interest motivated in no small part by a result of Heydar Radjavi [22]stating that any band of idempotents in a matrix ring over a field is (upper)triangularizable. (See also [8]–[10] and [20].) On the other hand, the studyof skew lattices (noncommutative variants of lattices - proper definition to

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follow) has focused attention on bands in rings that also possess join-likeoperations of the kind mentioned above. (See [4]–[7] and [15]–[18].)

By the Clifford-McLean Theorem, every band is a semilattice of rectan-gular bands. To be precise, given a band S, the Green’s equivalence D is acongruence on S with the quotient algebra S/D being the maximal semilat-tice image of S and each D-class being a maximal rectangular band in S.(Recall that eDf in a band if both efe = e and fef = f . Also a band isrectangular if efe = e holds.) These classes, considered as whole entities,have a semilattice ordering induced from S/D. More can be true for bandsin rings. Let R be a ring with an identity that satisfies the descending chaincondition on idempotents. Thus any descending chain e1 ≥ e2 ≥ ... in E(R)eventually stabilizes: en = en+1 = .... (Here ≥ denotes the natural partialordering of idempotents in any semigroup, given by e ≥ f if ef = f = fe.)In this case the D-classes of any band S in R such that 1 ∈ S (or S at leasthas a maximal D-class) are lattice ordered. In particular this is the case forbands in rings with identity that are finite dimensional algebras over fields.

The fact that bands in rings are often lattices of rectangular bands givesweight to the question of when a polynomial function j(x, y) can be definedon such a band S, such that for all e, f ∈ S, j(e, f) ∈ De ∨ Df , the joinD-class in S. The first obvious choice is the circle operation that works sowell when the idempotents commute. In general, however, even when twoidempotents e and f in a ring R have idempotent closure (ef and fe also liein E(R)), neither e ◦ f nor f ◦ e need be idempotent. Indeed, in any 3 × 3matrix ring over a field of characteristic 6= 2, an idempotent pair is given by

e =

0 0 00 1 10 0 0

and f =

0 1 00 1 00 0 0

.

While both ef and fe are idempotent, neither e ◦ f nor f ◦ e is such. But,given any ring R and any pair of idempotents e and f with idempotentclosure in R, if we instead use the cubic polynomial

e∇f = (e ◦ f)2 = e + f + fe − efe − fef

(with the second equality depending on idempotent closure) we obtain anidempotent that is further up in the ring than are e and f in that e(e∇f) = eand (e∇f)f = f hold. In general, given any subset S ⊆ E(R) for any ringR such that

S · S = {ef | e, f ∈ S} ⊆ E(R)

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we also haveS∇S = {e∇f | e, f ∈ S} ⊆ E(R).

In particular, S∇S ⊆ E(R) for any band S in a ring R. When such a bandis also closed under ∇, then it must have D-class joins since De ∨ Df is theD-class of e∇f . (These remarks, of course, hold even when R neither hasan identity nor satisfies the DCC on idempotents). In any case, ∇ servesas a cubic “noncommutative join” defined for all idempotent-closed pairs ofidempotents in a ring. Indeed, to extent that one sees multiplication as beingthe natural noncommutative meet, it would seem that correspondingly ∇ isthe natural noncommutative join. (A full study of the case for idempotent-closed pairs is carried out in [18].)

In this paper we study ∇-bands, that is, bands in rings that are closedunder ∇, viewing them as very well endowed bands, but also as noncom-mutative generalizations of lattices. It has five sections, the first of which isdevoted primarily to general properties of ∇-bands, but also contains somebackground on both bands and skew lattices. Sections 2 through 4 present anumber of necessary and sufficient conditions under which ∇ is associative,and the ∇-band is a skew lattice. These criteria involve the commutator[x, y] = xy − yx (Theorem 2.3), the Green’s relations L and R (Theorem3.3), associative joins on the entire band induced from sublattices (Theo-rems 3.4 and 3.5), the circle operation (Theorem 4.1) and finally, primitivesubalgebras (Theorem 4.2). In the fifth and final section we explore theproblem of when a band S in a ring will generate a ∇-band under · and ∇.

1 Skew lattices and ∇-bands

In this section we consider general properties of ∇-bands. Since ∇-bands are“possibly nonassociative” variants of distributive skew lattices we begin byrecalling some fundamental concepts and several ring-based examples thatare pertinent to our study of ∇-bands.

A skew lattice is a set S with a pair of associative, binary operations ∨and ∧ that together satisfy the following absorption identities:

x ∧ (x ∨ y) = x = (y ∨ x) ∧ x and x ∨ (x ∧ y) = x = (y ∧ x) ∨ x.

These identities force both ∨ and ∧ to be idempotent. A skew lattice S issymmetric if for all x, y ∈ S, x ∨ y = y ∨ x if and only if x ∧ y = y ∧ x.It is distributive if both x ∧ (y ∨ z) ∧ x = (x ∧ y ∧ x) ∨ (x ∧ z ∧ x) andx ∨ (y ∧ z) ∨ x = (x ∨ y ∨ x) ∧ (x ∨ z ∨ x) hold for all x, y, z ∈ S.

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Both band reducts (S,∨) and (S,∧) of a skew lattice S are regular inthat they satisfy the identity xyxzx = xyzx. But a skew lattice is regular ina stronger sense. The Green’s relation L(∧) defined for ∧ by xL(∧)y if bothx ∧ y = x and y ∧ x = y coincides with the Green’s relation R(∨) definedfor ∨ by xR(∨)y if both x ∨ y = y and y ∨ x = x. Similarly one has R(∧)

coinciding with L(∨). Setting L = L(∧) and R = R(∧), both relations arecongruences on S with S/L being the maximal right-handed skew latticeimage of S (satisfying x∧y∧x = y∧x and x∨y∨x = x∨y) and S/R beingthe maximal left-handed skew lattice image of S (satisfying x∧ y∧x = x∧ yand x ∨ y ∨ x = y ∨ x). Moreover, L ∩ R is the trivial congruence ∆ andL ∨ R = L ◦ R = R ◦ L is the Green’s congruence D given by xDy if bothx∧ y ∧x = x and y ∧x∧ y = y. Equivalently, xDy if both x∨ y ∨x = x andy∨x∨y = y. The induced image S/D is the maximal lattice image of S andthe various D-classes are maximal rectangular skew lattices in S. They arerectangular in that x∧y∧x = x = x∨y∨x holds for all x, y in any particularD-class. In addition, each D-class is characterized by x ∧ y = y ∨ x.

For further background on skew lattices, see [17]. For further backgroundon either bands or Green’s relations, consult [11], [12] or [21].

Thanks to Theorem 1.2 below, any ∇-band for which ∇ is associativeis a skew lattice in a ring. These skew lattices are always symmetric anddistributive. There are several classes of bands in rings whose exemplarsform skew lattices under · and ∇.

i) Any rectangular band S of idempotents (where xyx = x) in a ring is arectangular skew lattice under ∇ since x∇y = x+y +yx−x−y = yx.

ii) Every maximal left regular (xyx = xy) or right regular (xyx = yx)band in a ring is also closed under ◦. Thus ∇ = ◦ is trivially associa-tive. (See [15].) In general, any subset of idempotents in a ring R thatis closed under both · and ◦ is a ∇-band that forms a skew lattice.

iii) Every maximal normal band (xyzw = xzyw) in a ring is closed under∇ with ∇ being associative here also. (See [16].)

iv) Every band in a ring R on two generators e and f generates under ·and ∇ a skew lattice of ≤ 16 elements. (See [18] for further details.)

(v) On the other hand, ∇-bands exist for which ∇ is not associative. Thefollowing example with two D-classes is found in Cvetko-Vah’s disser-tation ([6] Example 49).

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S =

0 x1 x2 x1y1 + x2y2

0 1 0 y1

0 0 1 y2

0 0 0 0

∣

∣

∣

∣

∣

xi, yj ∈ C

∪

0 u uv1 uv2

0 1 v1 v2

0 0 0 00 0 0 0

∣

∣

∣

∣

∣

u, vi ∈ C

.

Upon setting

A =

0 0 0 00 1 0 00 0 1 00 0 0 0

, B =

0 1 0 00 1 0 00 0 0 00 0 0 0

and C =

0 0 0 00 1 1 00 0 0 00 0 0 0

we get,

A∇(B∇C) =

0 0 0 00 1 0 00 0 1 00 0 0 0

6=

0 0 −1 00 1 0 00 0 1 00 0 0 0

= (A∇B)∇C.

In the remainder of this section we establish some fundamental propertiesof ∇-bands. The natural partial order on a band, given by e ≥ f if ef =f = fe, refines the natural quasi-order given by e � f if fef = f . Theequivalence induced from � is the Green’s relation D. Turning to ∇-bandsproper, we say that a multiplicative congruence θ on a ∇-band is a ∇-congruence if θ is also a congruence under ∇. By Theorem 1.2(ii) below, Dis a ∇-congruence on any ∇-band, but this is not always the case for therefining equivalences, L and R. Before stating Theorem 1.2, we first recalla result from [4].

Lemma 1.1 Given a ∇-band S in a ring, for all a, b ∈ S:

i) a∇b = a + ba − aba and b∇a = a + ab − aba when b � a.

ii) (a∇b)∇a = a∇(b∇a) = a + b − bab.

Theorem 1.2 For any ∇-band S in a ring the following hold.

i) As a band, S is regular. That is, abaca = abca holds on S.

ii) D is a ∇-congruence and S/D is a lattice with Dx ∨Dy = Dx∇y.

iii) a(a∇b) = a = (b∇a)a and a∇(ab) = a = (ba)∇a.

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iv) ab = ba if and only if a∇b = b∇a.

v) a(b∇c)a = aba∇aca and a∇(bc)∇a = (a∇b∇a)(a∇c∇a).

Thus (S;∇, ·) is a distributive, symmetric skew lattice whenever ∇ is asso-ciative.

Proof. (i) Suppose A and B are D-classes in S with A > B. Let a ∈ A andb ∈ B so that a � b. Consider b∇a∇b = a + b − aba. From

(a + b − aba)a(a + b − aba) = a + b − aba and a(a + b − aba)a = a,

b∇a∇b ∈ A follows. On the other hand, clearly b∇a∇b ≥ b. Thus givenD-classes A > B in S, for all b ∈ B there exists a ∈ A such that a > b.Hence S is regular. (See [17] Theorem 1.11.) To see (ii), suppose that a � band a � c. Since S is regular, bab = b, bac = bc, cab = cb and cac = c.Hence (b∇c)a(b∇c) = b∇c so that a � b∇c. Thus the D-class of b∇c is thejoin-class of Db and Dc and D is a ∇-congruence. Next, (iii) follows fromsuch routine calculations as

a∇(ab) = a + ab + aba − aaba − abaab = a + ab + aba − aba − ab = a.

(iv) a∇b and b∇a only differ by the terms ba and ab. For the first part of(v), regularity gives

a(b∇c)a = a(b + c + cb − bcb − cbc)a= aba + aca + acaba − abacaba − acabaca = aba∇aca.

For the other identity, first observe that [b(c∇a)]2 = (bc + ba + bac− bcac−baca)2 expands as

(bc + bcba − bca) + (babc + ba − baca) + (babc + bacba − baca)−(bcabc + bcaba − bca) − (babc + bacba − baca)

= bc + ba − baca + babc − bcabc

which in turn must equal b(c∇a) = bc + ba + bac − bcac − baca. Equatingand cancelling common terms gives babc − bcabc = bac − bcac, that is, theidentity

babc + bcac = bac + bcabc. (∗)

Thus(a∇b∇a)(a∇c∇a) = (a + b − bab)(a + c − cac) = a + bc − bcac − babc + bac= a + bc − bac − bcabc + bac = a + bc − bcabc = a∇(bc)∇a.

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∇-bands have other properties first observed in skew lattices. The nextresult demonstrates the important role played by instances of commutativity.In particular it exhibits the effects that ∇-closure has on the multiplicativereduct (S, ·) of any ∇-band S.

Theorem 1.3 Join classes and meet classes are given by commuting joinsand commuting meets. That is, given D-classes A and B in a ∇-band S,their join D-class J and meet D-class M are

J = {a∇b | a ∈ A, b ∈ B&a∇b = b∇a} and M = {ab | a ∈ A, b ∈ B&ab = ba}.

Moreover, for every a ∈ A there exists b ∈ B such that a∇b = b∇a in J andab = ba in M .

Proof. Given v ∈ J , there exist a ∈ A and b ∈ B such that v ≥ a, b. (Seta = va′v and b = vb′v for any a′ ∈ A and b′ ∈ B.) For such a and b we havea∇b ∈ J and

v = v(a∇b)v = v(a + b + ba − aba − bab)v = a + b + ba − aba − bab = a∇b.

Similarly, b∇a equals v also and the assertion about J is seen. The case forM is similar. For the final assertion, pick a in A and let v ∈ J be such thatv ≥ a. That b ∈ B exists such that a∇b = v = b∇a is now clear. The restfollows from symmetry.

Corollary 1.4 Given a ∇-band S and e ∈ S, the following are equivalent:

i) De = {e}.

ii) For all x ∈ S, e∇x = x∇e and ex = xe.

Proof. Clearly (ii) implies (i). That the member of a trivial D-class com-mutes with all members of S follows from the final assertion of Theorem 1.3.

Corollary 1.5 Any set of commuting elements in a ∇-band S generates asublattice of S, that is, a subalgebra on which ∨ and ∧ are commutative,thus making it a sublattice.

We next consider various criteria for ∇ to be associative, thus making a∇-band a skew lattice.

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2 Associativity of ∇ and the commutator [x, y]

Given elements x, y in a ring R, recall that their commutator is [x, y] =xy − yx. Clearly x and y commute if and only if [x, y] = 0. For any pair ofidempotents e and f in a band S in R

e∇f − e ◦ f = ef + fe − efe − fef = [e, f ]2.

Thus [e, f ]2 = 0 on S if and only if ∇ and ◦ agree as binary operations on S.In this section we show that a ∇-band S is associative if and only if for alle, f ∈ S, [e, f ]2 lies in the center of the subring of R generated by S. Thatis, ∇ is associative if and only if g[e, f ]2 = [e, f ]2g for all e, f, g ∈ S. Thisfully generalizes a criterion of Cvetko-Vah given initially for a special casein [4]. We begin with a pair of lemmas.

Lemma 2.1 (a∇b)c(a∇b) = −(abca − aca − bca − bcb + bcab).

Proof. Multiplying out (a + b + ba − aba − bab)c(a + b + ba − aba − bab)and cancelling yields

(aca + acb − acab) + (bca + bcb − bcab) − (abca + abcb − abcab)= aca + bca + bcb − bcab − abca

where the underlined terms vanish collectively by (*).

Lemma 2.2 Given a, b, c in a ∇-band S, a∇(b∇c) = (a∇b)∇c if and onlyif

[b, c]2a − a[b, c]2a = c[a, b]2 − c[a, b]2c.

Proof.

a∇(b∇c) = a + b + c + cb − bcb − cbc + ba + ca + cba − bcba − cbca−aba − aca − acba + abcba + acbca − (b∇c)a(b∇c)

while

(a∇b)∇c = a + b + ba − aba − bab + c + ca + cb + cba − caba − cbab−cac − cbc − cbac + cabac + cbabc − (a∇b)c(a∇b).

Equating a∇(b∇c) with (a∇b)∇c and then cancelling yields

−bcb − cbc − bcba − cbca − aca − acba + abcba + acbca − (b∇c)a(b∇c)= −bab − caba − cbab − cac − cbc − cbac + cabac + cbabc − (a∇b)c(a∇b).

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Applying the previous lemma gives

−bcb − cbc − bcba − cbca − aca − acba + abcba+acbca + bcab − bab − cab − cac + cabc

= −bab − caba − cbab − cac − cbc − cbac + cabac+cbabc + abca − aca − bca − bcb + bcab

which reduces to

−bcba − cbca − acba + abcba + acbca − cab + cabc= −caba − cbab − cbac + cabac + cbabc + abca − bca.

Rearranging, grouping and factoring gives,

bca − bcba − cbca − a(bc − cb)2a = cab − caba − cbab − c(ab − ba)2c,

Adding cba to both sides, then grouping and factoring once again gives

(bc − cb)2a − a(bc − cb)2a = c(ab − ba)2 − c(ab − ba)2c

which is the statement of the lemma.

This leads to the main characterization of ∇-associativity for this section.

Theorem 2.3 A ∇-band S is associative if and only if a[b, c]2 = [b, c]2a forall a, b, c ∈ S.

Proof. This identity implies the identity of Lemma 2.2 making ∇ asso-ciative. On the other hand, replacing a by aba and b by bab in the latterand simplifying gives c(ab − ba)2c = c(ab − ba)2 and thus a(bc − cb)2a =(bc − cb)2a upon cancelling. The result follows after permuting variables inc(ab − ba)2c = c(ab − ba)2 to get a(bc − cb)2a = a(bc − cb)2.

Thus the associativity of ∇ reduces to a question of commutativity: does[x, y]2 always produce elements lying in the center of the subring S± gener-ated by S? This is another indication of the subtle role of commutativity inthe study of noncommutative lattices.

3 Associativity of ∇ and the Green’s equivalences

L and R

Given elements a, b in a band S, recall that aLb if ab = a and ba = b.Similarly, aRb if ab = b and ba = a. Both L and R refine D; moreover L is

9

a right congruence on S in that aLb implies acLbc for all c ∈ S. SimilarlyR is a left congruence on S. If S is regular, then both L and R are bandcongruences. In particular, L and R are multiplicative congruences on all∇-bands. We turn to the status of L and R as ∇-congruences on a ∇-band.But first let a∇b = b and b∇a = a be denoted by aR∇b and similarly leta∇b = a and b∇a = b be denoted by aL∇b.

Lemma 3.1 In a ∇-band, aLb if and only if aR∇b and similarly aRb ifand only if aL∇b. In general, aLb implies (c∇a)(c∇b) = c∇a for all c andaRb implies (a∇c)(b∇c) = b∇c for all c.

Proof. Expanding, a∇b = b and b∇a = a reduce to a = aba + bab − baand b = aba + bab − ab. Multiplying on the left by a and b respectively,yields a = ab and b = ba, that is aLb. Conversely, if aLb under the ringmultiplication, then a∇b reduces to b and b∇a reduces to a. In general, forall c ∈ S, (c∇a)(c∇b) = (c+a+ac−aca− cac)(c+ b+ bc− bcb− cbc) whichwith the assistance of the band congruence L expands as

c+(ac+a−acb)+(ac)−(ac+aca−acb)−(cac) = c+a+ac−aca−cac = c∇a

Similarly, (c∇b)(c∇a) = c∇b so that c∇aLc∇b. The case for R is similar.

In general L and R need not be ∇-congruences. However:

Lemma 3.2 L and R are ∇-congruences on a ∇-band S if and only if forall a, b, c ∈ S,

i) a(bc − cbc)a = a(bc − cbc).

ii) a(bc − bcb)a = (bc − bcb)a.

Proof. Given uLv, (u∇x)(v∇x) = (u + x + xu − uxu − xux)(v + x + xv −vxv − xvx) expands and simplifies to

(u) + (xv + x − xvx) + (xu) − (uxu) − (xu) = u + x + xv − uxu − xvx.

Thus, (u∇x)(v∇x) = u∇x holds for uLv iff xv−xvx = xu−xux. Replacingu and v by generic values, uvu and vu, we see that L is a ∇-congruenceprecisely if xuvu − xuvux = xvu − xvux for all u, x, v in S. Likewise, Ris a ∇-congruence if uvux − xuvux = uvx − xuvx holds for all u, x, v inS. Parts (i) and (ii) result from changing letters and rearranging terms inthese identities.

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Theorem 3.3 Given a ∇-band S in a ring, both L and R are ∇-congruencesif and only if ∇ is associative and thus S is a skew lattice.

Proof. If ∇ is associative so that S is a skew lattice, then both L and Rmust be full skew lattice congruences. On the other hand, given that L andR are ∇-congruences, the identities of the above lemma imply that

a[b, c]2 = a(bc − cbc + cb − bcb) = a(bc − cbc + cb − bcb)a= a(bc − bcb + cb − cbc)a = (bc − bcb + cb − cbc)a = [b, c]2a.

Thus, the criterion of Theorem 2.3 is satisfied and hence ∇ is associative.

A closely related criterion involves a canonical factorization that canoccur in ∇-bands. To begin, a sublattice of a ∇-band is a subalgebra onwhich ef = fe and hence e∇f = f∇e hold. Every sublattice T of a ∇-bandS meets each D-class in S in at most one element. A lattice section in Sis a sublattice T that meets each D-class exactly once. In this case T isan internal copy of S/D, the maximal lattice image of S, TR =

⋃

t∈T Rt

is a maximal right regular band in S and TL =⋃

t∈T Lt is a maximal leftregular band in S with TR and TL being copies of the maximal right andleft regular images S/L and S/R of S. Each e ∈ S factors as eLeR whereeL = ete ∈ TL, eR = tee ∈ TR where te is the unique element in T ∩ De.Put otherwise, eL and eR are the unique elements in De related to e and tein the following D-class picture.

e R eL = eteL L

tee = eR R te

Due to S being regular

ef = (eLeR)(fLfR) = (eLfL)(eRfR)

holds for all e, f ∈ S. Multiplication on S thus decomposes internally intoseparate products on TL and TR so that S is isomorphic to a sub-band ofTL × TR. We call the factorization e = eLeR the internal Kimura factor-ization of e relative to T . Even more is true.

Given a ∇-band S with lattice section T , a modified join operation ∨T

can be defined such that (S,∨T , ·) is a distributive, symmetric skew lattice.To begin, TL and TR are in fact skew lattices. Suppose that a, b ∈ TL.Certainly, a∇b ∈ S, but since a and b lie in the left regular band TL, wealso have a∇b = a ◦ b. Let ta, tb ∈ T be such that aLta and bLtb. By left

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regularity, we have abta = ab, atbta = atb, batb = ba and btatb = bta. Hence(a ◦ b)(ta ◦ tb), upon expansion, is easily seen to reduce to a ◦ b. Similarly,(ta ◦ tb)(a ◦ b) = ta ◦ tb. Thus a ◦ b ∈ TL, and TL is a skew lattice inS as claimed. Likewise, TR is a skew lattice in S. The internal Kimuradecomposition of the band S with respect to T enables us to define anoperation ∨T on S by setting

e ∨T f = (eL ◦ fL)(eR ◦ fR).

Clearly (e∨T f)L = eL ◦fL and (e∨T f)R = eR ◦fR. It follows that (S,∨T , ·)is an “internal” fibred product of the skew lattices (TL, ◦, ·) and (TR, ◦, ·)over their common sublattice T and thus is a skew lattice. We will call ∨T

the associative join on S induced from T . Clearly ∨T is dependent on Tand is thus somewhat more complex in design than ∇. Indeed, soon we willsee that in certain cases distinct lattice sections of S can produce differentassociative joins. But first we have:

Theorem 3.4 Given a ∇-band S with a lattice section T , the binary oper-ation ∇ is associative if and only if e∇f = e∨T f for all e, f ∈ S, where ∨T

is the T -induced associative join on S. Thus if ∇ is associative, all latticesections of S must induce a common associative join, namely ∇.

Proof. To begin, if ∇ equals ∨T , then clearly ∇ is associative. Conversely,let ∇ be associative. The regularity of ∇ and the fact that x∇y = yx in anyD-class of S gives,

e∇f = (e∇te∇f∇tf)∇(te∇e∇tf∇f) = (te∇e∇tf∇f)(e∇te∇f∇tf)= (ete∇ftf)(tee∇tff) = (eL ◦ fL)(eR ◦ fR) = e ∨T f.

The next-to-last equality is because ∇ reduces to ◦ on any left or rightregular band in S. The final assertion of the theorem is clear.

Consider the following pair of lattice sections, T and T ′, for Example (v)in Section 1, given respectively as follows:

0 0 0 00 1 0 00 0 1 00 0 0 0

>

0 0 0 00 1 0 00 0 0 00 0 0 0

and

0 0 0 00 1 0 00 0 1 00 0 0 0

>

0 0 0 00 1 1 00 0 0 00 0 0 0

.

If A and B as chosen in the example, then

A ∨T B =

0 1 0 00 1 0 00 0 1 00 0 0 0

and A ∨T ′ B =

0 1 1 00 1 0 00 0 1 00 0 0 0

.

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Hence the choice of T can affect the outcome of the induced join when ∇ isnot associative.

The above theorem and example raise the converse question of whetherhaving a unique associative join ∨ implies ∇ is associative. This is often thecase as the next theorem shows.

Theorem 3.5 If S is a ∇-band such that S/D is at most countable, then Shas a lattice section T . Such a lattice section can always be found to includeany given finite subset T0 of pairwise commuting elements in S. Finally, ifall lattice sections T of S induce a common associative join on S then ∇ isthis join and as such is associative.

Proof. Let D1,D2, ... be a complete listing of the D-classes of S. Pick ana1 ∈ D1 and consider the set X = {x ∈ S | a1x = xa1}. By Theorem 1.3,X is a band that contains a1 and X meets each D-class of S. Moreover, Xis a ∇-band. Indeed, given x, y ∈ X, then a1 and x + y + yx − xyx − yxyclearly commute so that x∇y ∈ X also. Note that D1 ∩ X = {a1}. Nextpick a2 ∈ D2 ∩ X, set Y = {y ∈ X|a2y = ya2}. Again Y is a ∇-bandthat meets each D-class of S and moreover Di ∩ X = {ai} for i = 1, 2.The process continues through the countable set of D-classes to produce thelattice section T = {a1, a2, a3, ...}. By placing the D-classes of the elementsin T0 at the front of the list of D-classes, the second assertion follows. Finally,suppose that all lattice sections T induce a common associative join ∨ onS. Let e, f ∈ S be given. Then e and f generate a skew lattice S0 in S withat most 16 elements. (See [18].) Take a lattice section T0 of S0 and extendit to a full lattice section T of S, using the first part of this theorem. Thene∨T f = e∨T0

f and the latter equals e∇f in S0 by Theorem 3.4. Thus e∇fis the common associative join e ∨ f induced by all lattice sections.

The Kimura decomposition in its external form was first given for regularbands in [14]. Its internal and less general version for skew lattices appearsin [6] Chapter 3 and [7]. While “really big”∇-bands need not have latticesections, thanks to Theorem 3.5 all ∇-bands in finite dimensional matrixrings over fields have lattice sections.

Before proceeding to the next section, we first give a useful localizedvariant of these results.

Corollary 3.6 Given a ∇-band S, ∇ is associative if and only if for alleDu and fDv situations in S with uv = vu,

e∇f = (eu ◦ fv)(ue ◦ vf). (3.6.1)

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Proof. Given any e, f, g ∈ S, since S/D is distributive the three elementsgenerate a subalgebra S′ of S with only finitely many D-classes. By Theorem3.5 any commuting subset T0 of S′ can be extended to a lattice section Tof S. By Theorem 3.4 each such S′ is a skew lattice, and hence S is a skewlattice, if and only if the above identity holds.

4 Associativity of ∇ and primitive subalgebras

Clearly ∇ is associative on a given ∇-band S precisely when it is associativeon every sub-∇-band on three generators. Thus ∇ is associative if andonly if it is thus on all sub-∇-bands generated from at most 3 D-classes, oralternatively, on all sub-∇-bands with at most 18 D-classes (18 being theorder of a free distributive lattice on 3 generators). Corollary 3.6 implies:

Proposition 4.1 ∇ is associative on S if and only if it is associative onall subalgebras generated from at most 2 D-classes, or alternatively, on allsubalgebras with at most 4 D-classes.

This also follows from a result taken from Cvetko-Vah’s dissertation. Itforms a rough dual to Corollary 3.6 in that instead of multiplying circleproducts in the join D-class off e and f as in the corollary, we considercircle products of ordinary products in the meet D-class of e and f . Butbefore stating Cvetko-Vah’s result we first recall an important property ofregular bands, namely that of middle absorption. If e, f � g in a regularband S (so that ege = e and fgf = f), then egf = ef also. Indeed,egf = egfegf = egefefgf = efef = ef . This characterizes regular bandsas bands and is used frequently in the proofs to follow, usually with anappeal to regularity.

Theorem 4.2 ([6]) Given a ∇-band S, the following are equivalent:

i) ∇ is associative.

ii) Given eDu, fDv with uv = vu in S, then (efe) ◦ (fef) = (evfe) ◦(feuf).

iii) Given e, f and efDw in S, then (efe) ◦ (fef) = (ewfe) ◦ (fewf).

Proof. Given e, u, f, v as in (ii), note that (evfe) ◦ (feuf) = (euvfe) ◦(feuvf) by regularity, where efDuv. Thus if (iii) holds, then (ii) follows.Conversely, suppose that (ii) holds and let w ∈ Def . Set u = w∇e∇w ∈ De

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and v = w∇f∇w ∈ Df . Then w < u and w < v so that uv = uvwuv = wand similarly vu = w. The regularity of multiplication and (ii) give

(ewfe) ◦ (fewf) = (evufe) ◦ (fevuf) = (evfe) ◦ (feuf) = (efe) ◦ (fef).

Thus (ii) and (iii) are seen to be equivalent. Next, assume ∇ is associativeand let e, f, u and v be as stated in (ii). Then (3.6.1) must hold: e∇f =(eu ◦ fv)(ue ◦ vf). Expanding (3.6.1) gives

e + f + fe − efe − fef = (eu + fv − eufv)(ue + vf − uevf)= e + euvf − evf + fvue + f − fuef − efve − euf + ef

or

fe − efe − fef = euvf − evf + fvue − fuef − efve − euf + ef

= fvue − fuef − efve,

since S is a regular band, so that ef = e(u∇v)f and thus the four underlinedterms cancel. Switch e and u with f and v and then multiply the equalityby −1 to obtain

efe + fef − ef = evfe + feuf − euvf

which is just (ii). Conversely, (ii) implies (3.6.1). Indeed,

e∇f = e + f + fe − efe − fef = e + f − (fef ◦ efe)= e + f − (fuef) ◦ (efve) [by (ii)]= e + f + fvue − fuef − efve + 0= e + f + fvue − fuef − efve + (ef − euf + euvf − evf)= (eu + fv − eufv)(ue + vf − uevf) = (eu ◦ fv)(ue ◦ vf).

Recall that a ∇-band or a skew lattice is primitive if it has exactlytwo (necessarily comparable) D-classes. More generally, a band S is totallyquasi-ordered if for all e, f ∈ S, either e � f or f � e. (Again, e � f iffef = f , or equivalently for ∇-bands, e∇f∇e = e.) These bands are ofinterest since Cvetko-Vah proved in [5] that every maximal, totally quasi-ordered regular band in a ring is a ∇- band. (See also Corollary 5.5, below.)For such ∇-bands, either Theorem 4.2 or Corollary 3.6 imply that to checkfor associativity of ∇ one need only check for associativity on its primitivesubalgebras. It is natural to ask if the “totally quasi-ordered” assumptioncan be removed from this observation? Put otherwise, if ∇ is associative on

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all primitive subalgebras of a ∇-band in a ring, must it be associative on theband? Or can a ∇-band S exist with exactly four D-classes, two of whichare mutually incomparable, such that ∇ is associative on all five maximalprimitive subalgebras but not on all of S? This leads us to:

Theorem 4.3 If ∇ is associative on all primitive subalgebras of a given∇-band S, then ∇ is associative on S, and conversely.

Proof. Granted the assumption, we need only to show that ∇ is associa-tive on subalgebras generated from two (possibly incomparable) D-classes.Given two D-classes A,B with meet class M and join class J , consider thesubalgebra S′ = A ∪ B ∪ M ∪ J . Let a lattice section of S′ be given byT = {a0, b0,m0, j0}. Take a ∈ A, b ∈ B. We show that a∇b = a ∨T b. Tobegin

a ∨T b = (aa0 + bb0 − aa0bb0)(a0a + b0b − a0ab0b)= a + am0b − ab0b + bm0a + b − bm0ab − abm0a − aa0b + ab= a + bm0a + b − bm0ab − abm0a

since aa0b+ab0b−am0b = aj0b = ab. Of course a∇b = a+b+ba−aba−bab.Denoting the difference (a ∨T b) − (a∇b) by ∆(a,b), we have

∆(a,b) = bm0a − bm0ab − abm0a − ba + aba + bab.

The associativity of ∇ on the primitive ∇-band M ∪ B implies that

∆(abm0a,b) = bm0a − bm0ab − abm0a − bam0a + abm0a + bab

= bm0a − bm0ab − bam0a + bab = 0.

Subtracting from ∆(a,b) yields the refinement

∆(a,b) = bam0a + aba − abm0a − ba.

Using the latter to calculate ∆(a,bam0b) in the associative context of M ∪ Agives:

0 = ∆(a,bam0b) = bam0a + aba − abm0a − ba = ∆(a,b).

The converse is trivial.

Since ∇ is associative on a ∇-band only if it is thus on all primitivesubalgebra, two of our earlier criteria are fine-tuned for the primitive caseas follows.

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Corollary 4.4 Given a primitive ∇-band P with D-classes A > B, thefollowing are equivalent:

i) ∇ is associative on S.

ii) For all a ∈ A and b, c ∈ B, a[b, c]2 = [b, c]2a.

iii) For all a ∈ A and b, c ∈ B, both

a(bc − cbc)a = a(bc − cbc) and a(bc − bcb)a = (bc − bcb)a.

Proof. That (i) implies (ii) and (iii) follows from Theorem 2.3, Lemma 3.2and Theorem 3.3. The identities of (ii) and (iii), in the unconditional form,conversely imply (i). Given a, b, c in P , the only case where the identitiesneed not hold in general, is the case where a in A and b, c in B. In allalternative cases these identities hold. We check the case where b ∈ A andc ∈ B. Here we have b � a, c so that xby = xy wherever x and y are eithera or c. Hence

a[b, c]2 = a[bc + cb − bcb − cbc] = ac + acb − acb − ac = 0,a(bc − cbc) = abc − acbc = ac − ac = 0 = a(bc − cbc)a,

and similarly [b, c]2a = 0 and (bc− bcb)a = 0 = a(bc− bcb)a. The alternativecases, b, c ∈ A or b ∈ B but c ∈ A, are similarly verified, as in the case wherea, b, c ∈ B.

Given the matrices A ≻ B,C from Cvetko-Vah’s example in Section 1,we have:

A[B,C]2 =

0 0 0 00 0 0 00 0 0 00 0 0 0

6=

0 0 1 00 0 0 00 0 0 00 0 0 0

= [B,C]2A, and

A(BC − BCB)A =

0 0 0 00 0 1 00 0 0 00 0 0 0

6=

0 0 1 00 0 1 00 0 0 00 0 0 0

= (BC − BCB)A.

By contrast, consider next the four cases of associative ∇-bands in Sec-tion 1. In the second case one has [b, c]2 = bc+ cb− bcb− cbc = 0 holding sothat ∇ = ◦ and associativity is guaranteed. The other three cases are (mul-tiplicatively) normal ∇-bands. This paper’s various criteria provide quickalternative proofs to the fact that ∇ is associative on such bands.

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Corollary 4.5 A normal ∇-band S is a skew lattice. (That is, ∇ must beassociative.)

Proof 1. The identity xyzw = xzyw implies a[bc + cb − bcb − cbc] = 0 =[bc + cb − bcb − cbc]a. Hence ∇ is associative by Theorem 2.3.Proof 2. Again, xyzw = xzyw implies that the criteria of Lemma 3.2 aresatisfied, since all equations reduce to 0 = 0.Proof 3. And finally, xyzw = xzyw implies efe = egfe and fef = fegffor all g ∈ Def , and hence (efe) ◦ (fef) = (egfe) ◦ (fegf). Thus ∇ isassociative by Theorem 4.2.

Not only are maximal normal bands in rings necessarily skew lattices,in fact they form skew Boolean algebras since their principal inner ideals⌈e⌉ = {f | f ≤ e} are Boolean lattices. Bignall and Spinks have shown thatskew Boolean algebras have fundamental connections to both binary andternary discriminator varieties, as well as to implicative BCK algebras andtheir noncommutative variants, implicative BCS-algebras. (See [1]–[3], [16],[19] and [23].)

5 The ∇-closure of a band and ∇-inductive condi-

tions

Not every regular band in a ring need generate a ∇-band in that ring,although cases exist where generation is guaranteed. (By contrast, everyregular band S is isomorphic to a regular band S′ in some ring R′ such thatS′ generates an associative ∇-band in R′.) When a ∇-band is generatedfrom a band S it is called the ∇-closure of S. Our next theorem describeswhat must occur at any stage in a successful generation of a ∇-band froma regular band in a ring.

Theorem 5.1 Given a regular band S in a ring R and elements e, f ∈ S,S ∪ {e∇f} generates a (possibly larger) regular band S′ in R if and only iffor all a, b, c ∈ S,

I) ea(e ◦ f)bf = eabf .

II) a(e − ef)abc(f − ef)c = a(e − ef)b(f − ef)c.

Proof. Observe first that since S is regular, ea(−ef)bf = ea(fe − efe −fef)bf , so that ea(e∇f)bf = ea(e ◦ f)bf . Thus if the semigroup S′ gener-ated from S and e∇f is a regular band, then since ea, bf � e∇f we have

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ea(e∇f)bf = eabf so that (I) must follow. In general, elements in thesemigroup S′ generated from S and e∇f look like

a0(e∇f)a1(e∇f)a2(e∇f)...an−1(e∇f)an

with a0, a1, ..., an ∈ S1 for n ≥ 1. (The n = 0 case is just a0 ∈ S.) For S′ tobe a regular band we need both

(e∇f)a(e∇f)b(e∇f) = (e∇f)ab(e∇f) (5.1.1)

and[a(e∇f)b(e∇f)c]2 = a(e∇f)b(e∇f)c (5.1.2)

or by (5.1.1)

a(e∇f)bcab(e∇f)c = a(e∇f)b(e∇f)c (5.1.2′)

to hold for all a, b, c ∈ S. Setting Λ = {e, f, fe,−efe,−fef}, we get

(e∇f)a(e∇f)b(e∇f) =∑

u,v∈Λ

ua(e∇f)bv.

Except for the two cases, ea(e∇f)bf and fa(e∇f)be, all of the ua(e∇f)bvterms reduce to uabv terms due to the regularity of S. For example,

efa(e∇f)bfef = efae(e∇f)bfef = efaebfef = efabfef.

Thus (5.1.1) reduces first to ea(e∇f)bf + fa(e∇f)be = eabf + fabe andthen to

ea(e ◦ f)bf + fa(f ◦ e)be = eabf + fabe. (5.1.1′)

Clearly (I) implies (5.1.1′) and when S′ is regular, (I) holds so that (5.1.1′)follows. Thus (5.1.1′) can be replaced by the stronger assertion, (I).

Next, using a...bcab...c = a...abc...c, (5.1.2′) can be rearranged as

a(e∇f)abc(e∇f)c = a(e∇f)b(e∇f)c. (5.1.2′′)

Expanding, gives

a(e + f + fe − efe − fef)abc(e + f + fe − efe − fef)c= a(e + f + fe − efe − fef)b(e + f + fe − efe − fef)c.

Using the identity aeabcec = aebec holding for all regular bands, multiplyingout the left side of (5.1.2′′) creates a number of terms that reduce to the

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corresponding terms on the right. Of course we have, aeabcec = aebec andafabcfc = afbfec, but also cases such as

afe(abc)fefc = afef(abc)fefc = afef(b)fefc = afe(b)fefc,

andaef(abc)fefc = aef(abc)efc = aef(b)efc = aef(b)fefc.

Where this matching fails, (5.1.2′′) first reduces to

a(e)abc(f + fe − efe − fef)c + a(fe − efe − fef)abc(f)c

= a(e)b(f + fe− efe − fef)c + a(fe − efe − fef)b(f)c.

Using regularity again on products involving the underlined terms, the abovereduces to

a(e)abc(f − ef)c − a(ef)abc(f)c = a(e)b(f − ef)c − a(ef)b(f)c

or

a(e)abc(f)c−a(e)abc(ef)c−a(ef)abc(f)c = a(e)b(f)c−a(e)b(ef)c−a(ef)b(f)c.

Upon adding the identity a(ef)abc(ef)c = a(ef)b(ef)c, we have reduced(5.1.2) to (II).

A successful generation of a ∇-band from a regular band requires thatat each stage of generation the new regular band S′ also satisfies (I) and (II)for all a, b, c, e, f ∈ S′. While (I) and (II) generally need not be passed tolater bands, nonetheless Theorem 5.1 does suggest a strategy for generating∇-bands.

To this end, let C denote a condition potentially satisfied by regularbands in rings. C is ∇-inductive if (i) any regular band S in a ring thatsatisfies C must satisfy (I) and (II), and (ii) for all e, f ∈ S, the regularband S′ generated from S ∪ {e∇f} also satisfies C. Clearly:

Theorem 5.2 Any regular band of idempotents in a ring that satisfies a∇-inductive condition C will generate a ∇-band (its ∇-closure) in that ring.

We consider three conditions already known to be ∇-inductive (in ourparlance). We give new proofs based on the concept of ∇-induction.

Corollary 5.3 Left [right] regularity (uvu = uv[= vu]) is a ∇-inductivecondition.

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Proof. Given left regularity it is easy to see that (I) and (II) hold so thatS ∪ {e∇f} generates a regular band S′. Since S is left regular, e∇f = e ◦ fand (e◦f)a(e◦f) = (e◦f)a holds for all a ∈ S. This forces S′ to also be leftregular. Indeed (e ◦ f)a(e ◦ f) = (e + f − ef)a(e + f − ef) must expand toeae + eaf − eaef + fae + faf − faef − efae− efaf + efaef which, thanksto S being left regular, simplifies to ea+eaf −eaf +fae+fa−fae−efa−efa + efa = (e ◦ f)a.

Corollary 5.4 Normality (uxyv = uyxv) is a ∇-inductive condition.

Proof. Again, given normality, it is easy to see that (I) and (II) hold so thatS ∪{e∇f} generates a regular band. But the larger band also lies inside thering S± generated from S, and hence it is also normal since S± inheritedthe identity uxyv = uyxv from S.

Recall that a band S is totally quasi-ordered if for all e, f ∈ S, eithere � f or f � e, that is, either efe = e or fef = f . The following result isdue to Cvetko-Vah ([5] or [6]).

Corollary 5.5 Being totally quasi-ordered is a ∇-inductive condition.

Proof. In verifying (I), we use the fact that ea(e ◦ f)bf multiplies out toeaebf + eafbf − eaefbf . There are two cases to consider.Case 1: the �-maximum is e or f . Say e. Then eaebf + eafbf − eaefbfreduces to eabf + eafbf − eafbf = eabf . The subcase for f is similar.Case 2: the �-maximum is a or b. Say a. Here eaebf + eafbf − eaefbfreduces to ebf + efbf − efbf = ebf = eabf . The subcase for b is similar.

For (II), first suppose that e � f . Then both sides of (II) reduce to 0.Indeed:

a(e − ef)abc(f − ef)c = a(e − ef)fabc(f − ef)c = a0abc(f − ef)c = 0

and

a(e − ef)b(f − ef)c = a(e − ef)fb(f − ef)c = a0b(f − ef)c = 0.

Likewise, both sides reduce to 0 when f � e, so that (II) follows. HenceS∪{e∇f} generates a regular band S′ in R. Is S′ also totally quasi-ordered?Suppose that e � f in S. (The f � e case is similar.) Here e∇f = f + fe−fef . Thus f(e∇f)f = f while (e∇f)f(e∇f) = f(e∇f) = f(f +fe−fef) =e∇f . Since e∇fDf in S′, every element u′ in S′ must be D-equivalent tosome element u in S. Since S is totally quasi-ordered, so is S′.

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Problem. Find a necessary and sufficient condition for any regular bandin any ring to generate a ∇-band in that ring. That is, find a universal∇-inductive condition.

References

[1] R. J. Bignall and J. Leech, Skew Boolean algebras and discrimina-tor varieties, Algebra Universalis 33 (1995), 387–398.

[2] R. J. Bignall and M. Spinks, Implicative BCS-algebra subreducts ofskew Boolean algebras, Sci. Math. Japonica 59 (2003), 629–638.

[3] R. J. Bignall and M. Spinks, On binary discriminator algebras, I:implicative BCS-algebrs, International Journal of Algebra and Compu-tation, to appear.

[4] Cvetko-Vah, K., Pure skew lattices in rings, Semigroup Forum 68

(2004), 268–279.

[5] Cvetko-Vah, K., Pure ∇-bands, Semigroup Forum 71 (2005), 93–101.

[6] Cvetko-Vah, K., Skew Lattices in Rings. Dissertation, University ofLjubljana, 2005.

[7] Cvetko-Vah, K., Internal decompositions of skew lattices, Communi-cations in Algebra 35 (2007), 243–247.

[8] R. Drnovsek, An irreducible semigroup of idempotents, Studia Math-ematica 125 (1997), 97–99.

[9] P. Fillmore, G. MacDonald, M. Radjabalipour, H. Radjavi,Towards a classification of maximal unicellular bands, Semigroup Fo-rum 49 (1994), 195–215.

[10] P. Fillmore G. MacDonald, M. Radjabalipour, H. Radjavi,Principal-ideal bands, Semigroup Forum 59 (1999), 362–373.

[11] Grillet, P. A., Semigroups, An Introduction to the Structure Theory,Marcel Dekker, New York, 1995.

[12] Howie, J. M., Fundamentals of Semigroup Theory, Clarendon Press,Oxford, 1995.

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[13] Jordan, P., Uber nichtkommutative Verbande, Arch. Math. 2 (1949),56–59.

[14] Kimura, N., The structure of idempotent semigroups, (I), Pacific Jour-nal of Mathematics 8 (1958), 257–275.

[15] Leech, J., Skew lattices in rings, Alg. Universalis 26 (1989), 48–72.

[16] Leech, J., Skew Boolean algebras, Alg. Universalis 27 (1990), 497–506

[17] Leech, J., Recent developments in the theory of skew lattices, Semi-group Forum 52 (1996), 7–24.

[18] Leech, J., Small skew lattices in rings, Semigroup Forum 70 (2005),307–311.

[19] Leech, J. and M. Spinks, Skew Boolean algebras derived from gener-alized Boolean algebras, Algebra Universalis, to appear.

[20] Livshits, L., G. MacDonald, B.Mathes, H. Radjavi, Reduciblesemigroups of idempotent operators, J. Operator Theory 40 (1998), 35–69.

[21] Petrich, M., Lectures on Semigroups, John Wiley and Sons, NewYork, 1977.

[22] H. Radjavi, On the reduction and triangularization of semigroups ofoperators, J. Operator Theory 13 (1985) 63–71.

[23] Spinks, M., Contributions to the Theory of Pre-BCK Algebras, Disser-tation, Monash University, 2002.

Department of Mathematics Department of MathematicsUniversity of Ljubljana Westmont CollegeJadranska 19 955 La Paz Road1000 Ljubljana, Slovenia Santa Barbara, CA 93105karin.cvetko@fmf.uni-lj.si USA

leech@westmont.edu

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