Dual binary discriminator varieties

Abstract

Left normal bands, strongly distributive skew lattices, implicative BCS-algebras, skew Boolean algebras, skew Boolean intersection algebras, and certain other non-commutative structures occur naturally as term reducts in the study of ternary discriminator algebras and the varieties that they generate, giving rise thereby to various classes of pointed discriminator varieties 1 that generalise the class of pointed ternary discriminator varieties. For each such class of varieties there is a corresponding pointed discriminator function that generalises the ternary discriminator. In this paper some of the classes of pointed discriminator varieties that are contained in the class of dual binary discriminator varieties are characterised. A key unifying property is that the principal ideals of an algebra in a dual binary discriminator variety are entirely determined by the dual binary discriminator term for that variety.

PBZ*-Lattices: Structure Theory and Subvarieties

“…Another subvariety of PBZL * where our description of ideals can be considerably simplified is the variety V (AOL) generated by all antiortholattices. Bignall and Spinks first observed that the variety of distributive BZ-lattices is a binary discriminator variety [4]. We extend their observation by noticing that V (AOL) is itself a binary discriminator variety.…”

“…at 0. Binary discriminator varieties are paramount among subtractive varieties with equationally definable principal ideals [1]; they were thoroughly studied in the unpublished [4].…”

“…Among the consequences of this remark we have a very slender description of V (AOL)-ideals. In fact, recall from [4] that if A is an algebra in a 0-binary discriminator variety V and b 0 (x, y) is the term witnessing this property for V, I ⊆ A is a V-ideal of A exactly when, for any a, c ∈ A, if c ∈ I and b A 0 (a, c) ∈ I, then a ∈ I. Therefore:…”

Ordinal and Horizontal Sums Constructing PBZ*-lattices

“…Note that, in any orthomodular lattice L, for all x ∈ L, x BZL,L = {0, x, x ′ , 1} ∈ {D 1 , D 2 , D 2 2 } ⊂ BA = V (D 2 ) ⊂ V (D 3 ) = V (D 4 ), where the last equality follows from the easy to notice facts that D 3 ∈ H BZL (D 4 ) and D 4 ∈ S BZL (D 2 ×D 3 ). If M is an antiortholattice and x ∈ M , then, clearly,…”

“…For a start, PBZ*-lattices can be seen as a common generalisation of orthomodular lattices [1] and of Kleene algebras [20] with an additional unary operation. In the lattice of subvarieties of PBZL * , moreover, we happen to encounter many situations of intrinsic interest in universal algebra: to name a few, subtractive varieties with equationally definable principal ideals that fail to be point-regular [2]; binary discriminator varieties [8,2]; ternary discriminator varieties generated by a single finite non-primal algebra.…”