Page 22 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
possessing a maximal element 1 becomes a Boolean algebra upon setting xʹ = 1 \ x. In particular,
every generalized Boolean algebra that is complete as a lattice forms a complete Boolean algebra
and thus satisfies all the identities of Theorem 1.1.15. Generalized Boolean algebras play a basic
role in the study of skew Boolean algebras in Chapter 5, being both the commutative cases of the
latter as well their maximal lattice images.
1.2 Bands
A band is a semigroup S whose elements are idempotent. Thus x2 = x (in multiplicative
notation) for all x in S. A band that is also commutative is called a semilattice. Clearly:
Lemma 1.2.1. If S is a commutative semigroup, then the set E(S) of all idempotents in S
forms a semilattice under the semigroup operation. £
When a semigroup S is not commutative, E(S) need not be closed under multiplication.
Closure of E(S) is obtained, however, with a weakened version of commutativity. A semigroup
is mid-commutative (or weakly commutative) if it satisfies the identity uxyv = uyxv. A mid-
commutative band is called a normal band. The next result is trivial.
Lemma 1.2.2. Given a mid-commutative semigroup S, the set of idempotents E(S) forms
a normal band under the given multiplication. £
Given a band S, several quasi-orders can be defined on S. To begin, the natural partial
order ≥ is defined on S by e ≥ f if ef = f = fe. The natural partial order refines the natural quasi-
order ≻ on S defined by e ≻ f if fef = f. Between ≥ and ≻ lie the left and right quasi-orders, ≻L
and ≻R defined by respectively by: e ≻L f if fe = f and e ≻R f if ef = f.
In the lattice of all quasi-orders on the underlying set of S, ≥ is the meet (intersection) of
≻L and ≻R, and ≻ is the join of ≻L and ≻R. That ≥ is a partial order, and that ≻L and ≻R are
quasi-orders meeting at ≥ are easily verified. To see that of ≻ is a quasi-order that is the join of
≻L and ≻R we will need the equivalences L = ≻L ∩ ≻Lop and R = ≻R ∩ ≻R op. Alternatively, L
and R are defined by: e L f if both ef = e and fe = f, and e R f if both ef = f and fe = e. L is a right
congruence (e L f ⇒ eg L fg for all g ∈ S) and R is a left congruence (e R f ⇒ ge L gf for all
g ∈ S). E.g., e L f implies egfg = efgfg = efg = eg and likewise fgeg = fg. We now state:
Lemma 1.2.3. For any band S, ≻L o ≻R = ≻R o ≻L = ≻ and the result is a quasi-order.
Proof. If e ≻L o ≻R f, then for some g, ge = g and gf = f. From this e ≻R ef ≻L f follows. That
e ≻R ef is clear. That ef ≻L f follows from (ef)f = ef and g(ef) = f. Thus ≻L o ≻R ⊆ ≻R o ≻L.
The reverse inclusion is shown in similar fashion. This commuting composition is thus a quasi-
order; moreover, it contains ≻. Indeed, from e ≻ f we obtain e ≻R ef ≻L fef = f, so that ≻ lies in ≻R
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possessing a maximal element 1 becomes a Boolean algebra upon setting xʹ = 1 \ x. In particular,
every generalized Boolean algebra that is complete as a lattice forms a complete Boolean algebra
and thus satisfies all the identities of Theorem 1.1.15. Generalized Boolean algebras play a basic
role in the study of skew Boolean algebras in Chapter 5, being both the commutative cases of the
latter as well their maximal lattice images.
1.2 Bands
A band is a semigroup S whose elements are idempotent. Thus x2 = x (in multiplicative
notation) for all x in S. A band that is also commutative is called a semilattice. Clearly:
Lemma 1.2.1. If S is a commutative semigroup, then the set E(S) of all idempotents in S
forms a semilattice under the semigroup operation. £
When a semigroup S is not commutative, E(S) need not be closed under multiplication.
Closure of E(S) is obtained, however, with a weakened version of commutativity. A semigroup
is mid-commutative (or weakly commutative) if it satisfies the identity uxyv = uyxv. A mid-
commutative band is called a normal band. The next result is trivial.
Lemma 1.2.2. Given a mid-commutative semigroup S, the set of idempotents E(S) forms
a normal band under the given multiplication. £
Given a band S, several quasi-orders can be defined on S. To begin, the natural partial
order ≥ is defined on S by e ≥ f if ef = f = fe. The natural partial order refines the natural quasi-
order ≻ on S defined by e ≻ f if fef = f. Between ≥ and ≻ lie the left and right quasi-orders, ≻L
and ≻R defined by respectively by: e ≻L f if fe = f and e ≻R f if ef = f.
In the lattice of all quasi-orders on the underlying set of S, ≥ is the meet (intersection) of
≻L and ≻R, and ≻ is the join of ≻L and ≻R. That ≥ is a partial order, and that ≻L and ≻R are
quasi-orders meeting at ≥ are easily verified. To see that of ≻ is a quasi-order that is the join of
≻L and ≻R we will need the equivalences L = ≻L ∩ ≻Lop and R = ≻R ∩ ≻R op. Alternatively, L
and R are defined by: e L f if both ef = e and fe = f, and e R f if both ef = f and fe = e. L is a right
congruence (e L f ⇒ eg L fg for all g ∈ S) and R is a left congruence (e R f ⇒ ge L gf for all
g ∈ S). E.g., e L f implies egfg = efgfg = efg = eg and likewise fgeg = fg. We now state:
Lemma 1.2.3. For any band S, ≻L o ≻R = ≻R o ≻L = ≻ and the result is a quasi-order.
Proof. If e ≻L o ≻R f, then for some g, ge = g and gf = f. From this e ≻R ef ≻L f follows. That
e ≻R ef is clear. That ef ≻L f follows from (ef)f = ef and g(ef) = f. Thus ≻L o ≻R ⊆ ≻R o ≻L.
The reverse inclusion is shown in similar fashion. This commuting composition is thus a quasi-
order; moreover, it contains ≻. Indeed, from e ≻ f we obtain e ≻R ef ≻L fef = f, so that ≻ lies in ≻R
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