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Identities for a class of regular unary semigroups. (English) Zbl 1157.20036

A semigroup \(S\) with an additional unary operation \(x\mapsto x^\circ\) is called a regular unary semigroup if \(S\) satisfies the identities (RU1) \(xx^\circ x=x\) and (RU2) \(x^\circ xx^\circ=x^\circ\). The author studies the following unary semigroup identities: (T) \((x_1^\circ\cdots x_n^\circ)^\circ=x_n^{\circ\circ}\cdots x_1^{\circ\circ}\), (LRB) \(xx^\circ yy^\circ xx^\circ=xx^\circ yy^\circ\), (RRB) \(x^\circ xy^\circ yx^\circ x=y^\circ yx^\circ x\). He shows that a regular unary semigroup \(S\) satisfies (T), (LRB) and (RRB) iff the set \(S^\circ=\{s^\circ\mid s\in S\}\) is an inverse subsemigroup containing a unique inverse of each element of \(S\) and gives various characterizations and properties of regular unary semigroups satisfying (LRB) and (RRB) or (LRB) and (T).
Reviewer’s remark. The author writes (p. 2501) that it is still open whether or not (RU1) is a corollary of (RU2), (T), (LRB) and (RRB). However, it is clear that (RU2), (T), (LRB) and (RRB) hold in the 2-element semilattice \(\{0,1\}\) with the unary operation \(x\mapsto x^\circ\) defined by \(x^\circ=0\) for all \(x\), while (RU1) fails.

MSC:

20M17 Regular semigroups
20M05 Free semigroups, generators and relations, word problems
20M10 General structure theory for semigroups
20M07 Varieties and pseudovarieties of semigroups
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