Page 45 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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II: Skew Lattices
skew lattice with a finite maximal lattice image must factor into a fibered product of primary
factors, the latter being algebras of rather simple type. In general, strongly distributive skew
lattices can be embedded in powers of a special primitive skew lattice 5, a noncommutative
5-element variant of the lattice 2 for which the latter is its maximal lattice image. (See Theorem
2.6.12.)
Finally, the material presented in this chapter first appeared in the papers referenced at
the end of this chapter.
2.1 Fundamental results
Any noncommutative lattice has two D-equivalences, D(∨) and D(∧), with each being a
congruence with respect to its associated operation. For skew lattices we have:
Lemma 2.1.1. Given a skew lattice, R(∨) = L(∧), L(∨) = R(∧), and D(∨) = D(∧). Thus the
common equivalence D is a congruence of skew lattices.
Proof. That R(∨) = L(∧) and L(∨) = R(∧) follow immediately from the above dualities. Hence:
D(∨) = L(∨)oR(∨) = R(∧)oL(∧) = D(∧). £
Thus for any skew lattice (S; ∨, ∧) we set R = R(∧) = L(∨) and L = L(∧) = R(∨). A skew
lattice is rectangular if either (S, ∨) or (S, ∧) is a rectangular band in which case, thanks to the
above dualities, both are rectangular bands with x∨y = y∧x. Put otherwise, a rectangular skew
lattice is precisely an antilattice for which R(∨) = L(∧) and L(∨) = R(∧).
Theorem 2.1.2. (The Clifford-McLean Theorem for skew lattices). Given a skew lattice
(S; ∨, ∧), the equivalence D is a congruence, S/D is the maximal lattice image of S and all
congruence classes of D are maximal rectangular skew lattices in S.
Proof. This follows from the above lemma and Corollary 1.3.6. £
Lemma 1.2.3, Theorem 1.3.3 and the above result give us:
Lemma 2.1.3. Given elements a and b in a skew lattice S,
a ≥(∧) b iff b ≥(∨) a and a ≻ (∧) b iff b ≻ (∨) a. £
Thus the natural partial order on any skew lattice is given by ≥ = ≥(∧) = ≤(∨) and the
natural quasiorder on any skew lattice is given by ≻ = ≻ (∧) = ≺ (∨).
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skew lattice with a finite maximal lattice image must factor into a fibered product of primary
factors, the latter being algebras of rather simple type. In general, strongly distributive skew
lattices can be embedded in powers of a special primitive skew lattice 5, a noncommutative
5-element variant of the lattice 2 for which the latter is its maximal lattice image. (See Theorem
2.6.12.)
Finally, the material presented in this chapter first appeared in the papers referenced at
the end of this chapter.
2.1 Fundamental results
Any noncommutative lattice has two D-equivalences, D(∨) and D(∧), with each being a
congruence with respect to its associated operation. For skew lattices we have:
Lemma 2.1.1. Given a skew lattice, R(∨) = L(∧), L(∨) = R(∧), and D(∨) = D(∧). Thus the
common equivalence D is a congruence of skew lattices.
Proof. That R(∨) = L(∧) and L(∨) = R(∧) follow immediately from the above dualities. Hence:
D(∨) = L(∨)oR(∨) = R(∧)oL(∧) = D(∧). £
Thus for any skew lattice (S; ∨, ∧) we set R = R(∧) = L(∨) and L = L(∧) = R(∨). A skew
lattice is rectangular if either (S, ∨) or (S, ∧) is a rectangular band in which case, thanks to the
above dualities, both are rectangular bands with x∨y = y∧x. Put otherwise, a rectangular skew
lattice is precisely an antilattice for which R(∨) = L(∧) and L(∨) = R(∧).
Theorem 2.1.2. (The Clifford-McLean Theorem for skew lattices). Given a skew lattice
(S; ∨, ∧), the equivalence D is a congruence, S/D is the maximal lattice image of S and all
congruence classes of D are maximal rectangular skew lattices in S.
Proof. This follows from the above lemma and Corollary 1.3.6. £
Lemma 1.2.3, Theorem 1.3.3 and the above result give us:
Lemma 2.1.3. Given elements a and b in a skew lattice S,
a ≥(∧) b iff b ≥(∨) a and a ≻ (∧) b iff b ≻ (∨) a. £
Thus the natural partial order on any skew lattice is given by ≥ = ≥(∧) = ≤(∨) and the
natural quasiorder on any skew lattice is given by ≻ = ≻ (∧) = ≺ (∨).
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