Subdirectly Irreducible Nonidempotent Left Distributive Left Quasigroups

Abstract

Abstract. Left distributive left quasigroups are binary algebras with unique left division satisfying the left distributive identity x(yz) ≈ (xy)(xz). In other words, binary algebras where all left translations are automorphisms. We provide a description and examples of non-idempotent subdirectly irreducible algebras in this class.

On the axioms of singquandles

“…Even if comparing quandles is as difficult as comparing links, as for the case of Vassiliev theory, the introduction of quandles (and racks) in knot theory paved the way for the construction of new invariants and techniques. Moreover, beside the interest of quandles for knot theory, these structures are relevant in many other areas, as theoretical physics, for the study of the Yang-Baxter equation (see [AG03,ESS99,ESG01]) or abstract algebra itself (see [Sta15,BS21,BF21]). In [BEHY18] and [CCE21] the singquandle construction is done for the oriented case, while in [NOS19] the notion of psyquandles is introduced for the case of pseudoknots and singular knots and links as a generalization of biquandle structures for classical and virtual links [FJSK04].…”

Subdirectly irreducible medial quandles

“…It is well-known that for so-called n-symmetric (and therefore for all finite) quandles, the operation \ need not be taken into consideration since it is defined by means of multiplication (see e.g. [19,Section 8.6] or [22]). In our previous paper [8], the operation \ was not taken into account, either.…”

“…It is also interesting to note that classification of SI racks (non-idempotent quandles) uses substantially different techniques, see [10,22].…”