Page 187 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 187
Further Topics in Skew Lattices
Proof. Consider the following identity or its dual:
x ∨ (y ∧ z ∧ u ∧ y) ∨ x = x ∨ (y ∧ u ∧ z ∧ y) ∨ x. (5.4.2)
Note that x∨y∨x ≥ x ∨ (y ∧ z ∧ u ∧ y) ∨ x & x ∨ (y ∧ u ∧ z ∧ y) ∨ x ≥ x by (1.1) and (1.6) with the
middle expressions being D-equivalent, since z ∧ u D u ∧ z. Hence, if a skew lattice S is strictly
categorical, then (5.4.2) holds by Theorem 5.4.8(iii). Conversely, let (5.4.2) hold in S and
suppose that a ≥ both b, bʹ ≥ c in S with b D bʹ. Assigning x → c, y → a, z → b∧bʹ and u → bʹ∧b
reduces (5.4.2) to b = b∧bʹ∧b = bʹ∧b∧bʹ = bʹ making S strictly categorical by Theorem 5.4.8(iii)
again. £
While distributive skew lattices are categorical, they need not be strictly categorical; but:
Theorem 5.4.10. A strictly categorical skew lattice S is distributive if and only if it is also quasi-
distributive.
Proof. Any distributive skew lattice is quasi-distributive. Conversely, in any strictly categorical
skew lattice both a ≥ a∧(b∨c)∧a and a ≥ (a∧b∧a) ∨ (a∧c∧a). In turn, a∧c∧b∧a ≤ both
a∧(b∨c)∧a and (a∧b∧a) ∨ (a∧c∧a). Indeed, regularity and absorption give, e.g.,
(a∧c∧b∧a)∧[a∧(b∨c)∧a] = a∧c∧b∧(b∨c)∧a = a∧c∧b∧a
and
(a∧c∧b∧a) ∧ [(a∧b∧a) ∨ (a∧c∧a)] = a∧c∧a∧b∧a ∧ [(a∧b∧a) ∨ (a∧c∧a)]
= a∧c∧a∧b∧a = a∧c∧b∧a
In any quasi-distributive skew lattice, a∧(b∨c)∧a D (a∧b∧a) ∨ (a∧c∧a). Thus if S is quasi-
distributive and strictly categorical, Theorem 5.4.8(iii) implies that both (5.2.1) and dually (5.2.2)
must hold. The converse is clear. £
Theorems 5.4.8 and 5.4.10 can also be used to show that a distributive, strictly
categorical skew lattice S is simply cancellative. It is (fully) cancellative when S is also
symmetric.
Corollary 5.4.11. A skew lattice is strictly categorical and distributive if and only if no
subalgebra is a copy of lattices M3 or N5 or either of the skew chains in Theorem 5.4.7(iv). £
Order-closure and paranormal skew lattice
Both normal skew lattices and conormal skew lattices are proper subvarieties of the
variety of strictly categorical skew lattices. It is reasonable to ask if these subvarieties jointly
generate the larger variety. This turns not to be the case since both types of algebras belong to
another variety of skew lattices that excludes many primitive skew lattices, all of which must be
strictly categorical.
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Proof. Consider the following identity or its dual:
x ∨ (y ∧ z ∧ u ∧ y) ∨ x = x ∨ (y ∧ u ∧ z ∧ y) ∨ x. (5.4.2)
Note that x∨y∨x ≥ x ∨ (y ∧ z ∧ u ∧ y) ∨ x & x ∨ (y ∧ u ∧ z ∧ y) ∨ x ≥ x by (1.1) and (1.6) with the
middle expressions being D-equivalent, since z ∧ u D u ∧ z. Hence, if a skew lattice S is strictly
categorical, then (5.4.2) holds by Theorem 5.4.8(iii). Conversely, let (5.4.2) hold in S and
suppose that a ≥ both b, bʹ ≥ c in S with b D bʹ. Assigning x → c, y → a, z → b∧bʹ and u → bʹ∧b
reduces (5.4.2) to b = b∧bʹ∧b = bʹ∧b∧bʹ = bʹ making S strictly categorical by Theorem 5.4.8(iii)
again. £
While distributive skew lattices are categorical, they need not be strictly categorical; but:
Theorem 5.4.10. A strictly categorical skew lattice S is distributive if and only if it is also quasi-
distributive.
Proof. Any distributive skew lattice is quasi-distributive. Conversely, in any strictly categorical
skew lattice both a ≥ a∧(b∨c)∧a and a ≥ (a∧b∧a) ∨ (a∧c∧a). In turn, a∧c∧b∧a ≤ both
a∧(b∨c)∧a and (a∧b∧a) ∨ (a∧c∧a). Indeed, regularity and absorption give, e.g.,
(a∧c∧b∧a)∧[a∧(b∨c)∧a] = a∧c∧b∧(b∨c)∧a = a∧c∧b∧a
and
(a∧c∧b∧a) ∧ [(a∧b∧a) ∨ (a∧c∧a)] = a∧c∧a∧b∧a ∧ [(a∧b∧a) ∨ (a∧c∧a)]
= a∧c∧a∧b∧a = a∧c∧b∧a
In any quasi-distributive skew lattice, a∧(b∨c)∧a D (a∧b∧a) ∨ (a∧c∧a). Thus if S is quasi-
distributive and strictly categorical, Theorem 5.4.8(iii) implies that both (5.2.1) and dually (5.2.2)
must hold. The converse is clear. £
Theorems 5.4.8 and 5.4.10 can also be used to show that a distributive, strictly
categorical skew lattice S is simply cancellative. It is (fully) cancellative when S is also
symmetric.
Corollary 5.4.11. A skew lattice is strictly categorical and distributive if and only if no
subalgebra is a copy of lattices M3 or N5 or either of the skew chains in Theorem 5.4.7(iv). £
Order-closure and paranormal skew lattice
Both normal skew lattices and conormal skew lattices are proper subvarieties of the
variety of strictly categorical skew lattices. It is reasonable to ask if these subvarieties jointly
generate the larger variety. This turns not to be the case since both types of algebras belong to
another variety of skew lattices that excludes many primitive skew lattices, all of which must be
strictly categorical.
185
