Romanowska, A.; Roszkowska, B. Representations of n-cyclic groupoids. (English) Zbl 0669.20058 Algebra Univers. 26, No. 1, 7-15 (1989). An n-cyclic groupoid is a groupoid satisfying the following four identities: \((xy)z=(xz)y\), \(xx=x\), \(x(yz)=xy\), \((((xy)y)...)y=x\) where y is repeated n times. A decomposition of an n-cyclic groupoid into a disjoint sum of abelian groups is found and the result is applied to describe free objects, to establish properties of congruence relations and to characterize subdirectly irreducible groupoids in the variety of n-cyclic groupoids. Reviewer: J.Ježek Cited in 8 Documents MSC: 20N99 Other generalizations of groups 08A05 Structure theory of algebraic structures Keywords:identities; disjoint sum of abelian groups; free objects; congruence relations; subdirectly irreducible groupoids; variety of n-cyclic groupoids PDFBibTeX XMLCite \textit{A. Romanowska} and \textit{B. Roszkowska}, Algebra Univers. 26, No. 1, 7--15 (1989; Zbl 0669.20058) Full Text: DOI References: [1] G. Gr?tzer,Universal Algebra, 2nd edition, Springer-Verlag, New York, 1979. [2] J. Je?ek andT. Kepka,Medial Groupoids, Rozpravy Ceskoslovensk? Akademie Ved 93, Academia, Praha, 1983. [3] A. I. Mal’cev,Multiplication of classes of algebraic systems (Russian), Sibirsk, Math. Z.8 (1967), 346-365. [4] J. P?onka,On algebras with n distinct essentially n-ary operations, Algebra Universalis1 (1971), 73-79. · Zbl 0219.08006 · doi:10.1007/BF02944958 [5] J. P?onka,On k-cyclic groupoids, Math. Japonica30 (1985), 371-382. · Zbl 0572.08004 [6] R. Quackenbush,Equational classes generated by finite algebras, Algebras Universalis1 (1971), 265-266. · Zbl 0231.08004 · doi:10.1007/BF02944989 [7] A.Romanowska and B.Roszkowska,On some groupoid modes, to appear in Demonstratio Math. · Zbl 0669.08005 [8] A. Romanowska, J. D. H. Smith,Modal Theory-an Algebraic Approach to Order, Geometry and Convexity, Helderman Verlag, Berlin, 1985. · Zbl 0553.08001 [9] B. M. Schein,Homomorphisms and subdirect decompositions of semigroups. Pacific J. Math17 (1966), 529-547. · Zbl 0197.01603 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.