On quasivarieties generated by (3-3)-snake lattices
DOI:
https://doi.org/10.31489/2026m3/49-60Keywords:
lattice, modular lattice, quasivariety, splitting pair, semi-splitting pair, profiniteness, finite basis of quasi-identities, non-standard topological quasivarietyAbstract
This paper addresses two classical and interrelated questions in universal algebra and lattice theory concerning the finite axiomatizability of quasivarieties and the standardness of topological quasivarieties generated by finite lattices. These problems have long attracted significant attention due to their central role in understanding the boundary between algebraic structure and logical definability. Although a general characterization remains elusive, specific families of finite lattices provide valuable insight into the complexity of quasi-equational theories. In this work, we investigate the quasivariety generated by the so-called (3-3)-snake lattice, a finite structure known for its intricate congruence behavior. We demonstrate that the quasivariety determined by this lattice is neither finitely axiomatizable nor standard in the topological sense. This result contributes to the broader program of identifying finite lattices whose quasi-identities inherently resist finite axiomatizability and whose associated Boolean topological algebras fail to exhibit profiniteness. In addition to establishing the main theorems, we present a generalization that expands the scope of the construction, offering an illustrative example and a corollary that highlight the robustness of the method. The findings underscore the continued importance of studying finite lattices with atypical quasi-equational properties and enrich the existing landscape of nonstandard quasivarieties.
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