Page 46 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 46
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Lemma 2.1.4. On any skew lattice S, both R and L are congruences. In particular, both
algebraic reducts (S, ∨) and (S, ∧) are regular bands and given e ≻ a, b ≻ f in S:
a ∨ f ∨ b = a ∨ b and a ∧ e ∧ b = a ∧ b.
Proof. Given a skew lattice S, we first show that (S, ∧) satisfies the class covering condition.
Indeed, given comparable D-classes A > B in S with b an arbitrary element in B, then for any a in
A, b ≥(∨) b∨a∨b ∈ A. Thus b∨a∨b ≥(∧) b and (S, ∧) is seen to satisfy the class covering
condition. Thus by Theorem 1.2.16, both L(∧) and R(∧) are ∧-congruences. Similarly, both L(∨)
and R(∨) are ∨-congruences. The first statement follows now from Lemma 2.1.1. Thus both
band reducts are regular; moreover, by Theorem 1.2.7, the dual conditional identities must hold.
£
A skew lattice for which D = R [D = L] is said to be right-handed [left-handed]. If S is
also rectangular, then it is right-rectangular [left-rectangular]. Remarks following Theorem
1.2.27 give us:
Theorem 2.1.5. (Kimura factorization for skew lattices). Given a skew lattice S, S/R is
the maximal left-handed image of S, S/L is the maximal right handed image of S and the
commuting diagram
S ⎯⎯⎯⎯ ⎯⎯⎯ ⎯⎯ ⎯ → S/L
⎜⎜
⎜⎜
⎜⎜
↓↓
S/R ⎯ ⎯ ⎯ ⎯⎯ ⎯ ⎯ ⎯ ⎯ → S/D
is a pullback diagram and S factors as the fibered product S ≅ S/R ×S/D S/L. £
An embedding of a band B into a skew lattice S is a semigroup embedding of B into the
band (S; ∧). Clearly for a band to be embedded into a skew lattice, the band must be regular. It
turns out that this condition is sufficient. Our easy proof of this fact brings us into the subject of
skew lattices in rings. We begin with the following considerations. Given a ring R with set of
idempotents E(R), a band in R is any subset of E(R) that is closed under multiplication. Clearly:
Proposition 2.1.6. Let R be a ring with identity 1 such that any descending chain
e1 ≥ e2 ≥ … in E(S) eventually stabilizes: en = en+1 =… for some n ≥ 1. Then the D-classes of
any band B in R such that 1 ∈ B are lattice-ordered. (This is the case for matrix rings over
fields.) £
44
Lemma 2.1.4. On any skew lattice S, both R and L are congruences. In particular, both
algebraic reducts (S, ∨) and (S, ∧) are regular bands and given e ≻ a, b ≻ f in S:
a ∨ f ∨ b = a ∨ b and a ∧ e ∧ b = a ∧ b.
Proof. Given a skew lattice S, we first show that (S, ∧) satisfies the class covering condition.
Indeed, given comparable D-classes A > B in S with b an arbitrary element in B, then for any a in
A, b ≥(∨) b∨a∨b ∈ A. Thus b∨a∨b ≥(∧) b and (S, ∧) is seen to satisfy the class covering
condition. Thus by Theorem 1.2.16, both L(∧) and R(∧) are ∧-congruences. Similarly, both L(∨)
and R(∨) are ∨-congruences. The first statement follows now from Lemma 2.1.1. Thus both
band reducts are regular; moreover, by Theorem 1.2.7, the dual conditional identities must hold.
£
A skew lattice for which D = R [D = L] is said to be right-handed [left-handed]. If S is
also rectangular, then it is right-rectangular [left-rectangular]. Remarks following Theorem
1.2.27 give us:
Theorem 2.1.5. (Kimura factorization for skew lattices). Given a skew lattice S, S/R is
the maximal left-handed image of S, S/L is the maximal right handed image of S and the
commuting diagram
S ⎯⎯⎯⎯ ⎯⎯⎯ ⎯⎯ ⎯ → S/L
⎜⎜
⎜⎜
⎜⎜
↓↓
S/R ⎯ ⎯ ⎯ ⎯⎯ ⎯ ⎯ ⎯ ⎯ → S/D
is a pullback diagram and S factors as the fibered product S ≅ S/R ×S/D S/L. £
An embedding of a band B into a skew lattice S is a semigroup embedding of B into the
band (S; ∧). Clearly for a band to be embedded into a skew lattice, the band must be regular. It
turns out that this condition is sufficient. Our easy proof of this fact brings us into the subject of
skew lattices in rings. We begin with the following considerations. Given a ring R with set of
idempotents E(R), a band in R is any subset of E(R) that is closed under multiplication. Clearly:
Proposition 2.1.6. Let R be a ring with identity 1 such that any descending chain
e1 ≥ e2 ≥ … in E(S) eventually stabilizes: en = en+1 =… for some n ≥ 1. Then the D-classes of
any band B in R such that 1 ∈ B are lattice-ordered. (This is the case for matrix rings over
fields.) £
44
