Page 107 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 107
III: Quasilattices, Paralattices and their Congruences
resulting operation from S3 ×T S4 back to S to get ∧*. Every refined quasilattice can in principle
be recovered from its derived skew lattice in this manner. When S1 ×T S2 and S3 ×T S4 are just
reducts of a common fibered factorization of (S, ∨, ∧) as a skew lattice, the algebra (S, ∨*, ∧*) is
also a skew lattice.
Note that results 3.4.7 – 3.4.9 above, are trivial consequences of the above theorem and
their skew lattice predecessors in Section 2.2.
3.5 The effects of the distributive identities
Connections between refined quasilattices and distributive properties exist which
particularly involve split quasilattices; however, distributive identities, much like absorption
identities, proliferate in the absence of commutativity.
To begin, a noncommutative lattice is fully distributive if it satisfies the identities:
D1: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). D2: (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c).
D3: a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). D4: (a ∧ b) ∨ c = (a ∧ c) ∨ (b ∧ c).
The four identities are mutually independent, unlike the case for lattices.
Remark. If one replaces (∨, ∧) by (+, •), these identities describe a semiring that is also
distributive in that addition distributes over multiplication. When both operations are idempotent,
one has an idempotent, distributive semiring or an ID-semiring. That is, ID-semirings are just
fully distributive double bands, but in ring notation. ID-semirings were introduced by Pastijn and
Romanowska [1982] and studied in several subsequent papers. For instance, varieties of ID-rings
where both operations are middle commutative (i.e., normal in band terminology) were classified
by Pastijn in [1983]. Two separate lines of research thus meet at fully distributive quasilattices.
We will consider connections between ID-semirings and fully distributive quasilattices after
Corollary 4.5.3 below.
A noncommutative lattice is bidistributive if it satisfies a slightly weaker pair of
identities:
D5: a ∧ (b ∨ c) ∧ d = (a ∧ b ∧ d) ∨ (a ∧ c∧ d).
D6: a ∨ (b ∧ c) ∨ d = (a ∨ b ∨ d) ∧ (a ∨ c∨ d).
Finally, a noncommutative lattice is distributive if it satisfies an even weaker pair of
identities:
D7: a ∧ (b ∨ c) ∧ a = (a ∧ b ∧ a) ∨ (a ∧ c∧ a).
D8: a ∨ (b ∧ c) ∨ a = (a ∨ b ∨ a) ∧ (a ∨ c∨ a).
Recall that all skew lattices in rings are distributive as are all skew Boolean algebras.
105
resulting operation from S3 ×T S4 back to S to get ∧*. Every refined quasilattice can in principle
be recovered from its derived skew lattice in this manner. When S1 ×T S2 and S3 ×T S4 are just
reducts of a common fibered factorization of (S, ∨, ∧) as a skew lattice, the algebra (S, ∨*, ∧*) is
also a skew lattice.
Note that results 3.4.7 – 3.4.9 above, are trivial consequences of the above theorem and
their skew lattice predecessors in Section 2.2.
3.5 The effects of the distributive identities
Connections between refined quasilattices and distributive properties exist which
particularly involve split quasilattices; however, distributive identities, much like absorption
identities, proliferate in the absence of commutativity.
To begin, a noncommutative lattice is fully distributive if it satisfies the identities:
D1: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). D2: (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c).
D3: a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). D4: (a ∧ b) ∨ c = (a ∧ c) ∨ (b ∧ c).
The four identities are mutually independent, unlike the case for lattices.
Remark. If one replaces (∨, ∧) by (+, •), these identities describe a semiring that is also
distributive in that addition distributes over multiplication. When both operations are idempotent,
one has an idempotent, distributive semiring or an ID-semiring. That is, ID-semirings are just
fully distributive double bands, but in ring notation. ID-semirings were introduced by Pastijn and
Romanowska [1982] and studied in several subsequent papers. For instance, varieties of ID-rings
where both operations are middle commutative (i.e., normal in band terminology) were classified
by Pastijn in [1983]. Two separate lines of research thus meet at fully distributive quasilattices.
We will consider connections between ID-semirings and fully distributive quasilattices after
Corollary 4.5.3 below.
A noncommutative lattice is bidistributive if it satisfies a slightly weaker pair of
identities:
D5: a ∧ (b ∨ c) ∧ d = (a ∧ b ∧ d) ∨ (a ∧ c∧ d).
D6: a ∨ (b ∧ c) ∨ d = (a ∨ b ∨ d) ∧ (a ∨ c∨ d).
Finally, a noncommutative lattice is distributive if it satisfies an even weaker pair of
identities:
D7: a ∧ (b ∨ c) ∧ a = (a ∧ b ∧ a) ∨ (a ∧ c∧ a).
D8: a ∨ (b ∧ c) ∨ a = (a ∨ b ∨ a) ∧ (a ∨ c∨ a).
Recall that all skew lattices in rings are distributive as are all skew Boolean algebras.
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