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Free right h-adequate semigroups. (English)
Semigroups, theory and applications, Proc. Conf., Oberwolfach/FRG 1986, Lect. Notes Math. 1320, 97-120 (1988).
[For the entire collection see Zbl 0635.00007.] Relation ${\cal L}\sp*$ on a semigroup S is defined by: a${\cal L}\sp*b$ iff a${\cal L}b$ in some oversemigroup of S (${\cal L}$ the Green relation). S is called right-adequate if every ${\cal L}\sp*$-class contains an idempotent and the idempotents commute. Denote for $a\in S$ this unique idempotent by $a\sp*$ (* is an additional idempotent unary operation on the semigroup). The subclass of right h-adequate semigroups can be characterized by identity $(xz)\sp*(yz)\sp*=(x\sp*y\sp*z\sp*)\sp*.$ In this paper are investigated free right h-adequate semigroups, which are shown to have many properties reminiscent to the properties of free inverse semigroups. As a consequence of the normal form construction for the elements it follows that the word problem is solvable. The Green relations are trivial. In the case of a finite set of generators the free h-adequate semigroup is hopfian. A subset Y generates a free *- subsemigroup iff Y is a suffix code etc.
Reviewer: Ja.Henno