Boolean-like algebras

Abstract

Using Vaggione's concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varietie

A New View of Effects in a Hilbert Space

“…The key observation motivating the introduction of Church algebras [37,41] is that many algebras arising in completely different fields of mathematics -including Heyting algebras, rings with unit, or combinatory algebras -have a term operation q satisfying the fundamental properties of the if-then-else connective: q (1, x, y) ≈ x and q (0, x, y) ≈ y. This motivates the next definition.…”

On noncommutative generalisations of Boolean algebras

“…Directly indecomposable algebras play an important role in the characterisation of the structure of a variety of algebras. For example, if the class of indecomposable algebras in a Church variety (see Section 3.1 and [16]) is universal, then any algebra in the variety is a weak Boolean product of directly indecomposable algebras. In this section we summarize the basic ingredients of factorisation: tuples of complementary factor congruences and decomposition operators (see [15]) .…”

“…, e n ) is the pure reduct of A. Church algebras of dimension 2 were introduced as Church algebras in [14] and studied in [16]. Examples of Church algebras of dimension 2 are Boolean algebras (with q(x, y, z) = (x ∧ y) ∨ (¬x ∧ z)) or rings with unit (with q(x, y, z) = xy + z − xz).…”

“…Boolean algebras are the main example of a well-behaved double-pointed variety -meaning a variety V whose type includes two distinct constants 0, 1 in every nontrivial A ∈ V. Since there are other double-pointed varieties of algebras that have Boolean-like features, in [16,10] the notion of Boolean-like algebra (of dimension 2) was introduced as a generalisation of Boolean algebras to a double-pointed but otherwise arbitrary similarity type. The idea behind this approach was that a Boolean-like algebra of dimension 2 is an algebra A such that every a ∈ A is 2-central in the sense of Vaggione [19], meaning that θ(a, 0) and θ(a, 1) are complementary factor congruences of A.…”

Factor Varieties and Symbolic Computation

“…(Boolean Algebras = Classical logic with a unary relation R satisfying ¬R(0) ∧ R(1) ∧ ∀x(¬R(x) → x = 0) ∧ ∀x(R(x) → x = 1)) Following Salibra et al [22], the factor variety axiomatized by fR(0, ξ f , ξt) = ξ f , fR(1, ξ f , ξt) = ξt, fR(x, ξ f , 1) = fR(x, ξ f , x) and fR(x, 0, ξt) = fR(x, x, ξt), is term equivalent to the variety of Boolean algebras. Up to isomorphism, we have only one factor algebra which corresponds to the Boolean algebra of truth values 2.…”