Page 263 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 263
VII: FURTHER TOPICS IN SKEW BOOLEAN ALGEBRAS
In Chapter IV the implicit perspective of a skew Boolean algebra is that of a skew lattice
with added structure, namely a constant 0 and a difference operation \, which with ∧ and ∨ satisfy
certain identities. There is an alternative approach: to consider algebras with a reduced signature
(∧, \, 0) or even (\, 0) that are subject to a set of identities, and view skew Boolean algebras as
instances where these simpler algebras acquire added structure. These reduced algebras can,
however, be of independent interest, as was the case for the iBCK algebras encountered below.
This chapter begins by looking at algebras of signature (\, 0) that satisfy “subtractive”
identities such as x\0 = x and x\x = 0 = 0\x. Indeed Section 7.1 considers seven such identities.
The first six (indeed the first four) characterize implicative BCS-algebras (or just iBCS-algebras).
The significance of these algebras lies in the fact that (\, 0)-reducts of skew Boolean algebras are
iBCS-algebras. The seventh identity, when joined to the rest, characterizes implicative BCK-
algebras (or just iBCK-algebras). A skew Boolean algebra is a generalized Boolean algebra if
and only if its (\, 0)-reduct is an iBCK-algebra. A skew Boolean algebra is simultaneously both a
strongly distributive skew lattice and an iBCS-algebra, having both types of algebras as reducts.
The Signature Bisection Theorem (Theorem 7.1.1) tells how a strongly distributive skew lattice
and an iBCS-algebra, if defined on a common set, must interact to form a skew Boolean algebra
having the initial pair of algebras as reducts. We next define an alternative iBCK-difference / on
skew Boolean ∩-algebras by x/y = x \ x∩y, in which case the reduct (S; /, 0) is an iBCK-algebra.
Theorem 7.1.2 characterizes skew Boolean ∩-algebras as algebras of signature (∨, ∧, /, 0).
The section continues by looking at the role of discriminator terms and discriminator
varieties. Skew Boolean algebras form a binary discriminator variety (Theorem 7.1.4) and all
binary discriminator algebras have an iBCS-algebra reduct; what is more, all iBCS-algebras also
have left-normal-band-with-0 reducts. Thus much, but not all, of the structure of a left-handed
skew Boolean algebra is encoded in its iBCS reduct (S; /, 0). The main result (Theorem 7.1.9)
states: If V is a binary discriminator variety with constant term 0 and additive term x + y, then
every algebra A of V has a left-handed skew Boolean algebra term reduct AS. (Here x + y is a
binary term satisfying x + 0 = x = 0 + x.) Clearly a right-handed version also holds. The section
concludes by looking at ternary discriminator varieties. Theorems 7.1.10 and 7.1.11 reveal a
close connection between skew Boolean ∩-algebras and pointed ternary discriminator varieties.
Most of Section 7.1 is based on the work of Robert Bignall and his student Matthew
Spinks, with some input from Jonathan Leech.
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In Chapter IV the implicit perspective of a skew Boolean algebra is that of a skew lattice
with added structure, namely a constant 0 and a difference operation \, which with ∧ and ∨ satisfy
certain identities. There is an alternative approach: to consider algebras with a reduced signature
(∧, \, 0) or even (\, 0) that are subject to a set of identities, and view skew Boolean algebras as
instances where these simpler algebras acquire added structure. These reduced algebras can,
however, be of independent interest, as was the case for the iBCK algebras encountered below.
This chapter begins by looking at algebras of signature (\, 0) that satisfy “subtractive”
identities such as x\0 = x and x\x = 0 = 0\x. Indeed Section 7.1 considers seven such identities.
The first six (indeed the first four) characterize implicative BCS-algebras (or just iBCS-algebras).
The significance of these algebras lies in the fact that (\, 0)-reducts of skew Boolean algebras are
iBCS-algebras. The seventh identity, when joined to the rest, characterizes implicative BCK-
algebras (or just iBCK-algebras). A skew Boolean algebra is a generalized Boolean algebra if
and only if its (\, 0)-reduct is an iBCK-algebra. A skew Boolean algebra is simultaneously both a
strongly distributive skew lattice and an iBCS-algebra, having both types of algebras as reducts.
The Signature Bisection Theorem (Theorem 7.1.1) tells how a strongly distributive skew lattice
and an iBCS-algebra, if defined on a common set, must interact to form a skew Boolean algebra
having the initial pair of algebras as reducts. We next define an alternative iBCK-difference / on
skew Boolean ∩-algebras by x/y = x \ x∩y, in which case the reduct (S; /, 0) is an iBCK-algebra.
Theorem 7.1.2 characterizes skew Boolean ∩-algebras as algebras of signature (∨, ∧, /, 0).
The section continues by looking at the role of discriminator terms and discriminator
varieties. Skew Boolean algebras form a binary discriminator variety (Theorem 7.1.4) and all
binary discriminator algebras have an iBCS-algebra reduct; what is more, all iBCS-algebras also
have left-normal-band-with-0 reducts. Thus much, but not all, of the structure of a left-handed
skew Boolean algebra is encoded in its iBCS reduct (S; /, 0). The main result (Theorem 7.1.9)
states: If V is a binary discriminator variety with constant term 0 and additive term x + y, then
every algebra A of V has a left-handed skew Boolean algebra term reduct AS. (Here x + y is a
binary term satisfying x + 0 = x = 0 + x.) Clearly a right-handed version also holds. The section
concludes by looking at ternary discriminator varieties. Theorems 7.1.10 and 7.1.11 reveal a
close connection between skew Boolean ∩-algebras and pointed ternary discriminator varieties.
Most of Section 7.1 is based on the work of Robert Bignall and his student Matthew
Spinks, with some input from Jonathan Leech.
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