Right product quasigroups and loops.
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Publication:2888139
zbMATH Open1253.20061arXiv0903.5436MaRDI QIDQ2888139
Aleksandar KrapeΕΎ, J. D. Phillips, Michael K. Kinyon
Publication date: 30 May 2012
Published in: Quasigroups and Related Systems (Search for Journal in Brave)
Abstract: Right groups are direct products of right zero semigroups and groups and they play a significant role in the semilattice decomposition theory of semigroups. Right groups can be characterized as associative right quasigroups (magmas in which left translations are bijective). If we do not assume associativity we get right quasigroups which are not necessarily representable as direct products of right zero semigroups and quasigroups. To obtain such a representation, we need stronger assumptions which lead us to the notion of emph{right product quasigroup}. If the quasigroup component is a (one-sided) loop, then we have a emph{right product (left, right) loop}. We find a system of identities which axiomatizes right product quasigroups, and use this to find axiom systems for right product (left, right) loops; in fact, we can obtain each of the latter by adjoining just one appropriate axiom to the right product quasigroup axiom system. We derive other properties of right product quasigroups and loops, and conclude by showing that the axioms for right product quasigroups are independent.
Full work available at URL: https://arxiv.org/abs/0903.5436
word problemaxiomatizationaxiom independencereproductive equationaxiom systemsright product loopsright product quasigroups
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