Affiliation:
1. Departamento de Matemática , Universidad Nacional del Sur INMABB – CONICET , Alem , 1253, Bahía Blanca , Argentina
2. Department of Mathematics , State University of New York , New Paltz , New York, 12561 , U.S.A
Abstract
Abstract
In 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A, →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (𝓘-zroupoid, for short) if A satisfies: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′ := x → 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids and A ∈ 𝓘. For x, y ∈ A, let x ∧ y := (x → y′)′ and x ∨ y := (x′ ∧ y′)′. In an earlier paper, we had proved that if A ∈ 𝓘, then the algebra A
mj
= 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 𝓘, the bisemigroup A
mj
is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity: x ∧ y ≈ x ∨ y, satisfies the Whitman Property. We conclude the paper with two open problems.
Reference26 articles.
1. Balbes, R.—Dwinger, P.: Distributive Lattices, University of Missouri Press, Columbia, 1974.
2. Bernstein, B. A.: A set of four postulates for Boolean algebras in terms of the implicative operation, Trans. Amer. Math. Soc. 36 (1934), 876–884.
3. Birkhoff, G.: Lattice Theory. 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, 1948.
4. Burris, S.—c, H. P.: A Course in Universal Algebra, Springer-Verlag, New York, 1981. The free version (2012) is available online as a PDF file at math.uwaterloo.ca/$\sim$snburris.
5. c, J. M.—Sankappanavar, H. P.: Order in implication zroupoids, Studia Logica 104 (2016), 417–453.