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Algebraic properties of the shift mapping. (English) Zbl 0679.20058
A J-algebra S is a semigroup (written here multiplicatively) together with another binary operation, say *, such that the following identities are satisfied: \(xy=(x*y)x\), \(x*(yz)=(x*y)(x*z)\) and \((xy)*z=x*(y*z)\). In this case, the groupoid S(*) is left selfdistributive. Interesting examples of J-algebras come from the following construction: Let F(A) be the monoid of injective transformations of a non-empty set A. The operation * is defined by \((f*g)(a)=fgf^{-1}(a)\) for \(a\in f(A)\) and \((f*g)(a)=a\) for \(a\not\in f(A)\). Then F(A) is a left cancellative J- algebra satisfying the identity \(x*((x*x)*x)=((x*x)*x)*(x*x)\) which is not satisfied in every J-algebra (and consequently, the subalgebra J generated in F(N) by the shift mapping \(n\to n+1\) is not free). Moreover, one-generated subalgebras of F(A) are only of the following three types: J, L, \(J\times L\), where L is a finite or a countably infinite semigroup of left units.
Reviewer: T.Kepka

MSC:
20M20 Semigroups of transformations, relations, partitions, etc.
17A30 Nonassociative algebras satisfying other identities
08A30 Subalgebras, congruence relations
17A50 Free nonassociative algebras
08B20 Free algebras
20N05 Loops, quasigroups
08A05 Structure theory of algebraic structures
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References:
[1] Patrick Dehornoy, Infinite products in monoids, Semigroup Forum 34 (1986), no. 1, 21 – 68. · Zbl 0601.20061 · doi:10.1007/BF02573152 · doi.org
[2] Patrick Dehornoy, \Pi \textonesuperior \(_{1}\)-complete families of elementary sequences, Ann. Pure Appl. Logic 38 (1988), no. 3, 257 – 287. · Zbl 0646.03030 · doi:10.1016/0168-0072(88)90028-0 · doi.org
[3] Patrick Dehornoy, Free distributive groupoids, J. Pure Appl. Algebra 61 (1989), no. 2, 123 – 146. · Zbl 0686.20041 · doi:10.1016/0022-4049(89)90009-1 · doi.org
[4] Randall Dougherty, A note on critical points of elementary embeddings, Notes. · Zbl 0791.03028
[5] Józef Dudek, Some remarks on distributive groupoids, Czechoslovak Math. J. 31(106) (1981), no. 3, 451 – 456. With a loose Russian summary. · Zbl 0472.20025
[6] David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37 – 65. · Zbl 0474.57003 · doi:10.1016/0022-4049(82)90077-9 · doi.org
[7] David Joyce, Simple quandles, J. Algebra 79 (1982), no. 2, 307 – 318. · Zbl 0514.20018 · doi:10.1016/0021-8693(82)90305-2 · doi.org
[8] T. Kepka, Notes on left-distributive groupoids, Acta Univ. Carolin. — Math. Phys. 22 (1981), no. 2, 23 – 37 (English, with Russian and Czech summaries). · Zbl 0517.20048
[9] Nobuo Nobusawa, A remark on conjugacy classes in simple groups, Osaka J. Math. 18 (1981), no. 3, 749 – 754. · Zbl 0473.20014
[10] R. S. Pierce, Symmetric groupoids, Osaka J. Math. 15 (1978), no. 1, 51 – 76. · Zbl 0395.20033
[11] Sherman K. Stein, Left-distributive quasi-groups, Proc. Amer. Math. Soc. 10 (1959), 577 – 578. · Zbl 0093.01902
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