Page 121 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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IV: SKEW BOOLEAN ALGEBRAS
In this chapter such fundamental concepts as normal bands, skew lattices and generalized
Boolean algebras are integrated. We have already seen in Theorem 2.3.7 that maximal normal
bands in rings form strongly distributive skew lattices that are characterized by
a∧(b ∨ c) = (a∧b) ∨ (a∧c) and (a ∨ b)∧c = (a∧c) ∨ (b∧c).
More is true: any such maximal normal band S forms a noncommutative variant of a generalized
Boolean algebra called a skew Boolean algebra. In particular S possesses a zero element 0 for
which 0∨a = a = a∨0 and 0∧a = 0 = a∧0 hold; S is also closed under a difference operation a \ b
given by a – aba. Such a system (S; ∨, ∧, \, 0) satisfies many identities, a subset of which
provides a defining set of identities for a skew Boolean algebra.
The strongly distributive skew lattices of partial functions encountered Section 2.6 give
archetypal examples: to ∨ and ∧ we adjoin the empty partial function ∅ as the zero and the
difference f \ g given by the restriction of f to F \ F∩G, where F and G are the respective supports
of f and g (their sets of actual inputs). In this chapter we will work with the right-handed case
PR(A, B). The skew Boolean version (with the expanded signature) is denoted by P(A, B).
In addition to the above classes of examples, every primitive skew lattice with a singleton
lower class A > {0} is strongly distributive and has a zero element 0; moreover a difference
operation is given by the simple rule: x \ y = x, if y = 0, and 0 otherwise. These primitive algebras
play a basic role in the theory, doing for skew Boolean algebras what the Boolean algebra 2 and
its isomorphic copes do for (generalized) Boolean algebras.
In the Section 4.1, skew Boolean algebras (S; ∨, ∧, \, 0) are formally defined as structural
enhancements of strongly distributive skew lattices. Variants of some of the familiar results
about generalized Boolean algebras are then proved. In particular, skew Boolean algebras are
shown to be subdirect products of primitive skew Boolean algebras; moreover every skew
Boolean algebra can be embedded into a power of 5, a 5-element primitive algebra. (See
Corollaries 4.1.6 and 4.1.7.) Not surprisingly, every right-handed skew Boolean algebra can be
embedded in some partial function algebra P(A, B).
In Section 4.2, special attention is given to classifying finite algebras, and in particular, to
classifying finitely generated (and thus finite) free skew Boolean algebras. In the process, not
only do we look at an important class of examples, we also engage in some of the basic algebraic
procedures of skew Boolean algebras. A fundamental concept in this section is that of an
orthosum, both an orthosum of elements and an orthosum of subalgebras. The main results are
Theorems 4.2.2 and 4.2.6, with the latter describing the structure of finitely generated free
algebras.
A skew Boolean algebra is, of course, just a strongly distributive skew lattice with added
operations and axiomatic constraints. While section 4.3 focuses mostly on the relation between
119
In this chapter such fundamental concepts as normal bands, skew lattices and generalized
Boolean algebras are integrated. We have already seen in Theorem 2.3.7 that maximal normal
bands in rings form strongly distributive skew lattices that are characterized by
a∧(b ∨ c) = (a∧b) ∨ (a∧c) and (a ∨ b)∧c = (a∧c) ∨ (b∧c).
More is true: any such maximal normal band S forms a noncommutative variant of a generalized
Boolean algebra called a skew Boolean algebra. In particular S possesses a zero element 0 for
which 0∨a = a = a∨0 and 0∧a = 0 = a∧0 hold; S is also closed under a difference operation a \ b
given by a – aba. Such a system (S; ∨, ∧, \, 0) satisfies many identities, a subset of which
provides a defining set of identities for a skew Boolean algebra.
The strongly distributive skew lattices of partial functions encountered Section 2.6 give
archetypal examples: to ∨ and ∧ we adjoin the empty partial function ∅ as the zero and the
difference f \ g given by the restriction of f to F \ F∩G, where F and G are the respective supports
of f and g (their sets of actual inputs). In this chapter we will work with the right-handed case
PR(A, B). The skew Boolean version (with the expanded signature) is denoted by P(A, B).
In addition to the above classes of examples, every primitive skew lattice with a singleton
lower class A > {0} is strongly distributive and has a zero element 0; moreover a difference
operation is given by the simple rule: x \ y = x, if y = 0, and 0 otherwise. These primitive algebras
play a basic role in the theory, doing for skew Boolean algebras what the Boolean algebra 2 and
its isomorphic copes do for (generalized) Boolean algebras.
In the Section 4.1, skew Boolean algebras (S; ∨, ∧, \, 0) are formally defined as structural
enhancements of strongly distributive skew lattices. Variants of some of the familiar results
about generalized Boolean algebras are then proved. In particular, skew Boolean algebras are
shown to be subdirect products of primitive skew Boolean algebras; moreover every skew
Boolean algebra can be embedded into a power of 5, a 5-element primitive algebra. (See
Corollaries 4.1.6 and 4.1.7.) Not surprisingly, every right-handed skew Boolean algebra can be
embedded in some partial function algebra P(A, B).
In Section 4.2, special attention is given to classifying finite algebras, and in particular, to
classifying finitely generated (and thus finite) free skew Boolean algebras. In the process, not
only do we look at an important class of examples, we also engage in some of the basic algebraic
procedures of skew Boolean algebras. A fundamental concept in this section is that of an
orthosum, both an orthosum of elements and an orthosum of subalgebras. The main results are
Theorems 4.2.2 and 4.2.6, with the latter describing the structure of finitely generated free
algebras.
A skew Boolean algebra is, of course, just a strongly distributive skew lattice with added
operations and axiomatic constraints. While section 4.3 focuses mostly on the relation between
119
