The connection of skew Boolean algebras and discriminator varieties to Church algebras

On noncommutative generalisations of Boolean algebras

“…Then if x ∈ e i /θ i a , we have x = t i (a, x, e i ) = a ∧ i x, therefore by definition of i r we conclude x i r a. The following proposition is a consequence of [4], Proposition 4.15, by observing that t i (a, x, y) = x for every x, y ∈ e i /θ i a . Proposition 7.2.…”

“…The skew reducts of a nBA. In [4] it is shown that the variety of skew BAs is term equivalent to the variety of SRCAs, whose type contains only a ternary operator. Here we use the n + 1-ary operator q of a nBA A to define ternary operators t 1 , .…”

“…The map x → x (rk) is bijective because (x (rk) ) (rk) = x by Lemma 4.8 (6) and (B4). The map is a homomorphism: t r (x, y, z) (rk) = L.4.8 (7) t r (x, y (rk) , z (rk) ) = t r ((x (rk) ) (rk) , y (rk) , z (rk) ) = q((x (rk) ) (rk) , y (rk) /r, z (rk) /r) = L.4.8 (4) q(x (rk) , y (rk) /k, z (rk) /k) = t k (x (rk) , y (rk) , z (rk) ). Moreover, (e r ) (rk) = e (rk)r = e k .…”

“…[4, Proposition 3.9] Let A = (A, q, 0) be an RCA. Every semicentral element e ∈ A determines a pair of complementary factor congruences:φ e = {(x,y) : q(e, x, y) = x} andφ e = {(x, y) : q(e, x, y) = y} such that φ e = θ(e, 0), the least congruence of A equating e and 0.…”

Factor Varieties and Symbolic Computation

“…Let R be a unary relation and 0 be a constant. Following Cvetko-Vah and Salibra [8], the factor variety axiomatized by fR(0, ξ f , ξt) = ξt and fR(x, ξ f , 0) = fR(x, ξ f , x), is term equivalent to the variety of right-handed skew Boolean algebras. A factor algebra A in this variety satisfies fR(0, ξ f , ξt) = ξt and fR(x, ξ f , ξt) = ξ f for all x ∈ A \ {0}.…”