Symmetric implication zroupoids and weak associative laws

Abstract

An algebra $${\mathbf {A}} = \langle A, \rightarrow , 0 \rangle $$ , where $$\rightarrow $$ is binary and 0 is a constant, is called an implication zroupoid ( $${\mathcal {I}}$$ -zroupoid, for short) if $${\mathbf {A}}$$ satisfies the identities: $$(x \rightarrow y) \rightarrow z \approx ((z' \rightarrow x) \rightarrow (y \rightarrow z)')'$$ and $$ 0'' \approx 0$$ , where $$x' := x \rightarrow 0$$ . An implication zroupoid is symmetric if it satisfies: $$x'' \approx x$$ and $$(x \rightarrow y')' \approx (y \rightarrow x')'$$ . The variety of symmetric $${\mathcal {I}}$$ -zroupoids is denoted by $${{\mathcal {S}}}$$ . We began a systematic analysis of weak associative laws (or identities) of length $$\le 4$$ 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 $${{\mathcal {S}}}$$ . In this paper, we complete the analysis by investigating the rest of the weak associative laws of length $$\le 4$$ relative to $${{\mathcal {S}}}$$ . We show that, of the (possible) 155 subvarieties of $${{\mathcal {S}}}$$ defined by the weak associative laws of length $$\le 4$$ , there are exactly 6 distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of $${{\mathcal {S}}}$$ defined by weak associative laws of length $$\le 4$$ .