Page 122 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 122
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
skew Boolean algebras and strongly distributive skew lattices, this is also a convenient context to
consider alternative characterizations of skew Boolean algebras. Theorem 4.3.5 gives an
independent set of five identities that characterizes all right-handed skew Boolean algebras.
In the Section 4.4 we look at skew Boolean algebras with finite intersections ∩, that is,
algebras for which the natural partial order ≥ has meets that are called intersections and denoted
by x∩y. For a partial function algebra P(A, B), f∩g is the usual intersection of partial functions f
and g viewed as subsets of A × B. All skew Boolean algebras S for which S/D is finite have
intersections as do, more generally, all complete skew Boolean algebras. Indeed, having at least
finite intersections is often the rule. Here, similarities with Boolean algebras are much tighter: if
∩ is included in the signature, then all congruences are determined by their kernel ideals – the
congruence classes of 0 – and thus their congruence lattices are distributive (Theorem 4.4.8). We
also show that free skew Boolean algebras have finite intersections (Theorem 4.4.18). The lattice
of all subvarieties of these algebras is described in the section’s concluding Theorem 4.4.24.
Sections 4.1 – 4.4 give a structural hierarchy lying at the core of this chapter’s subject.
skew Boolean algebras
with finite intersections
skew Boolean algebras
strongly distributive
skew lattices
In Section 4.5 we study a functor ω that object-wise takes a generalized Boolean algebra
B and constructs a skew Boolean cover of B, that is, a skew Boolean algebra SB such that (1)
SB/D ≅ B and (2) SB has a trivial center {0}. Its underlying set ω(B) consists of all naturally
ordered pairs in B, {(b, bʹ)⎪b ≥ bʹ}. Initially, ω(B) has obvious join and meet operations:
(b, bʹ) ∨ (c, cʹ) = (b∨c, bʹ∨cʹ) and (b, bʹ) ∧ (c, cʹ) = (b∧c, bʹ∧cʹ).
We “twist” these outcomes to produce noncommutative variants of ∨ and ∧. This provides
another class of skew Boolean algebras that have finite intersections as well as other properties
that we investigate. The process also gives us a “workout” of much that was discussed in
previous sections.
In Chapter 6, where we return to skew lattices in rings, special attention is given to skew
Boolean algebras that occur when the set of idempotents in a ring is closed under multiplication,
in which case the idempotents form such an algebra. In particular we will examine how this
occurrence affects the structure of the ring. All this occurs in Sections 6.4 to 6.7.
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skew Boolean algebras and strongly distributive skew lattices, this is also a convenient context to
consider alternative characterizations of skew Boolean algebras. Theorem 4.3.5 gives an
independent set of five identities that characterizes all right-handed skew Boolean algebras.
In the Section 4.4 we look at skew Boolean algebras with finite intersections ∩, that is,
algebras for which the natural partial order ≥ has meets that are called intersections and denoted
by x∩y. For a partial function algebra P(A, B), f∩g is the usual intersection of partial functions f
and g viewed as subsets of A × B. All skew Boolean algebras S for which S/D is finite have
intersections as do, more generally, all complete skew Boolean algebras. Indeed, having at least
finite intersections is often the rule. Here, similarities with Boolean algebras are much tighter: if
∩ is included in the signature, then all congruences are determined by their kernel ideals – the
congruence classes of 0 – and thus their congruence lattices are distributive (Theorem 4.4.8). We
also show that free skew Boolean algebras have finite intersections (Theorem 4.4.18). The lattice
of all subvarieties of these algebras is described in the section’s concluding Theorem 4.4.24.
Sections 4.1 – 4.4 give a structural hierarchy lying at the core of this chapter’s subject.
skew Boolean algebras
with finite intersections
skew Boolean algebras
strongly distributive
skew lattices
In Section 4.5 we study a functor ω that object-wise takes a generalized Boolean algebra
B and constructs a skew Boolean cover of B, that is, a skew Boolean algebra SB such that (1)
SB/D ≅ B and (2) SB has a trivial center {0}. Its underlying set ω(B) consists of all naturally
ordered pairs in B, {(b, bʹ)⎪b ≥ bʹ}. Initially, ω(B) has obvious join and meet operations:
(b, bʹ) ∨ (c, cʹ) = (b∨c, bʹ∨cʹ) and (b, bʹ) ∧ (c, cʹ) = (b∧c, bʹ∧cʹ).
We “twist” these outcomes to produce noncommutative variants of ∨ and ∧. This provides
another class of skew Boolean algebras that have finite intersections as well as other properties
that we investigate. The process also gives us a “workout” of much that was discussed in
previous sections.
In Chapter 6, where we return to skew lattices in rings, special attention is given to skew
Boolean algebras that occur when the set of idempotents in a ring is closed under multiplication,
in which case the idempotents form such an algebra. In particular we will examine how this
occurrence affects the structure of the ring. All this occurs in Sections 6.4 to 6.7.
120
