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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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