Symmetric implication zroupoids and weak associative laws.

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. 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 . [ABSTRACT FROM AUTHOR]