Symmetric implication zroupoids and weak associative laws

Abstract

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 the identities: (x→y)→z≈((z′→x)→(y→z)′)′ and 0 ′ ′≈ 0 , where x′: = x→ 0. An implication zroupoid is symmetric if it satisfies: x′ ′≈ x and (x→y′)′≈(y→x′)′. The variety of symmetric I-zroupoids is denoted by S. We began a systematic analysis of weak associative laws (or identities) of length ≤ 4 in Cornejo and Sankappanavar (Soft Comput 22(13):4319–4333, 2018a. https://doi.org/10.1007/s00500-017-2869-z), by examining the identities of Bol–Moufang type, in the context of the variety S. In this paper, we complete the analysis by investigating the rest of the weak associative laws of length ≤ 4 relative to S. We show that, of the (possible) 155 subvarieties of S defined by the weak associative laws of length ≤ 4 , there are exactly 6 distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of S defined by weak associative laws of length ≤ 4.