Page 23 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 23
I: Preliminaries
o ≻L. On the other hand ≻R o ≻L ⊆ ≻. For suppose e ≻R f ≻L g. But then f R fe and thus
g = gf R gfe = ge. It follows that geg = g so that e ≻ g. £
L and R were originally defined for arbitrary semigroups by J. A. Green, who showed
that LoR = RoL, with the resulting composition also being an equivalence denoted by D. (The
proof in the case of bands is similar to that for ≻L and ≻R.) Alternatively, D = ≻ ∩ ≻op. In the
lattice of all equivalences on the underlying set of S, D is the join of R and L, while the meet
R∩L is the equality relation Δ. To understand the role of these equivalences in the structure of
bands we turn to the class of bands that form the “anti-semilattices” amongst bands.
A rectangular band is a band S satisfying the identity xyx = x, or equivalently, the
identity xyz = xz. (Given the former identity, xyz = xyzxz = xz.) Rectangular bands thus form a
variety of bands for which xy = yx iff x = y. Examples are left zero bands having the
multiplication xy = x and right zero bands having the dual multiplication xy = y. Given a left
zero band L and a right zero band R, their direct product L × R is also a rectangular band. The
generality of such an example is demonstrated as follows.
Theorem 1.2.4. Given a rectangular band S with e ∈ S, L = {se ⎜s ∈ S} is a maximal
left zero band in S and R = {es ⎜s ∈ S} is a maximal right zero band in S; moreover the map
µ: L × R → S given by µ(x, y) = xy is an isomorphism. Finally, L is the L-class of e, R is the R-
class of e and S forms a single D-class.
Proof. Fixing e, since S is rectangular, sete = se so that L is at least a left zero semigroup in S. If
Lʹ were any left zero band in S such that e ∈ Lʹ, then for all s ∈ Lʹ, s = se so that s ∈L. Thus
Lʹ ⊆ L and L is indeed a maximal left zero band in S. Clearly such a subset must be an L-class of
S. Similarly, R is a stated. Finally, consider function µ: L × R → S. Viewing L × R as a direct
product of bands and applying the defining identity of a rectangular band we get:
µ((re, es)(te, eu)) = µ((rete, eseu)) = µ(re, eu) = reu = ru = resteu = µ(re, es)µ(te, eu)
so that µ is a homomorphism. Since s = (se)(es) for all s ∈S, µ maps L × R onto S. Finally, let
s = µ(ue, ev) = uev. Then se = ueve = ue and es = euev = ev so that (ue, ev) = (se, es). Thus µ
one-to-one also and an isomorphism. £
A rectangular band can be represented as a rectangular array with R-equivalent elements
comprising the rows and L-equivalent elements comprising the columns.
⎡e f g⎤
⎢⎥
⎣⎢ i j k ⎦⎥
21
o ≻L. On the other hand ≻R o ≻L ⊆ ≻. For suppose e ≻R f ≻L g. But then f R fe and thus
g = gf R gfe = ge. It follows that geg = g so that e ≻ g. £
L and R were originally defined for arbitrary semigroups by J. A. Green, who showed
that LoR = RoL, with the resulting composition also being an equivalence denoted by D. (The
proof in the case of bands is similar to that for ≻L and ≻R.) Alternatively, D = ≻ ∩ ≻op. In the
lattice of all equivalences on the underlying set of S, D is the join of R and L, while the meet
R∩L is the equality relation Δ. To understand the role of these equivalences in the structure of
bands we turn to the class of bands that form the “anti-semilattices” amongst bands.
A rectangular band is a band S satisfying the identity xyx = x, or equivalently, the
identity xyz = xz. (Given the former identity, xyz = xyzxz = xz.) Rectangular bands thus form a
variety of bands for which xy = yx iff x = y. Examples are left zero bands having the
multiplication xy = x and right zero bands having the dual multiplication xy = y. Given a left
zero band L and a right zero band R, their direct product L × R is also a rectangular band. The
generality of such an example is demonstrated as follows.
Theorem 1.2.4. Given a rectangular band S with e ∈ S, L = {se ⎜s ∈ S} is a maximal
left zero band in S and R = {es ⎜s ∈ S} is a maximal right zero band in S; moreover the map
µ: L × R → S given by µ(x, y) = xy is an isomorphism. Finally, L is the L-class of e, R is the R-
class of e and S forms a single D-class.
Proof. Fixing e, since S is rectangular, sete = se so that L is at least a left zero semigroup in S. If
Lʹ were any left zero band in S such that e ∈ Lʹ, then for all s ∈ Lʹ, s = se so that s ∈L. Thus
Lʹ ⊆ L and L is indeed a maximal left zero band in S. Clearly such a subset must be an L-class of
S. Similarly, R is a stated. Finally, consider function µ: L × R → S. Viewing L × R as a direct
product of bands and applying the defining identity of a rectangular band we get:
µ((re, es)(te, eu)) = µ((rete, eseu)) = µ(re, eu) = reu = ru = resteu = µ(re, es)µ(te, eu)
so that µ is a homomorphism. Since s = (se)(es) for all s ∈S, µ maps L × R onto S. Finally, let
s = µ(ue, ev) = uev. Then se = ueve = ue and es = euev = ev so that (ue, ev) = (se, es). Thus µ
one-to-one also and an isomorphism. £
A rectangular band can be represented as a rectangular array with R-equivalent elements
comprising the rows and L-equivalent elements comprising the columns.
⎡e f g⎤
⎢⎥
⎣⎢ i j k ⎦⎥
21
