**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXV, 1 (2006)

p. 1 - 19

Onesided inverses for semigroups

M. Petrich

**Abstract**.
In any semigroup *S*, we say that elements *a* and *b* are left
inverses of each other if *a = aba*, *b=bab* and *a L b*, in
which case we write *a g b*. Right inverses are defined dually with
the notation d. Set t = bg . We study the classes of
semigroups in which t has some of the usual properties of a
relation. We also consider properties of (maximal) completely simple
subsemigroups of *S*.

In terms of the above concepts, we characterize *E*-solid, central,
(almost) L -unipotent and locally (almost) L -unipotent
semigroups in many ways. We define these notions for arbitrary
semigroups by extending their definitions from regular semigroups.

**Keywords**:
Semigroup, inverse, onesided
inverse, *e*-solid, *E*-solid, idempotent square, central,
completely regular, almost q-idempotent, locally almost
q-idempotent, regular.

**AMS Subject classification:** Primary 20M99; Secondary 20M17.

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