Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.
We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present a new characterization of connected quandles that are affine.2000 Mathematics Subject Classification. Primary: 57M27. Secondary: 20N02, 20B10. Key words and phrases. Quandle, connected quandle, homogeneous quandle, affine quandle, enumeration of quandles, quandle envelope, transitive group of degree 2p. results on connected quandles in a simpler and faster way. We focus on enumeration of "small" connected quandles, namely those of order less than 48 (see Section 8 and Algorithm 8.1) and those with p or 2p elements (see Section 9). Our proof of non-existence of connected quandles with 2p elements, for any prime p > 5, is based on a new group-theoretical result for transitive groups of degree 2p, Theorem 10.1.The modern theory of quandles originated with Joyce's paper [24] and the introduction of the knot quandle, a complete invariant of oriented knots. Subsequently, quandles have been used as the basis of various knot invariants [4,5,6] and in algorithms on knot recognition [6,14].But the roots of quandle theory are much older, going back to self-distributive quasigroups, or latin quandles in today's terminology, see [38] for a comprehensive survey of results on latin quandles and their relation to the modern theory. Another vein of results has been motivated by the abstract properties of reflections on differentiable manifolds [27,30], resulting in what is now called involutory quandles [39]. Yet another source of historical examples is furnished by conjugation in groups, which eventually led to the discovery of the above-mentioned knot quandle by Joyce and Matveev [24,31].
Abstract. We provide a new characterization of several Mal'tsev conditions for locally finite varieties using hereditary term properties. We show a particular example how lack of absorption causes collapse in the Mal'tsev hierarchy, and point out a connection between solvability and lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal'tsev on blocks of every solvable congruence in every finite algebra in the variety.
Abstract. Using the Freese-McKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discuss the commutator of two normal subloops.Consequently, we argue that some standard definitions of loop theory, such as elementwise commutators and associators, should be revised and linked more closely to inner mappings. Using the new definitions, we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory. For instance, we show that the subloop generated by the new associators defined here is automatically normal. We conclude with a preliminary discussion of abelianess and solvability in loops.
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