## Niovi Kehayopulu and Michael Tsingelis

# On ordered left groups

## (Lobachevskii Journal of Mathematics, Vol.18, pp.131-137 )

Our purpose is to give some similarities and some
differences concerning the left groups between semigroups and
ordered semigroups. Unlike in semigroups (without order) if an
ordered semigroup is left simple and right cancellative, then it
is not isomorphic to a direct product of a zero ordered semigroup
and an ordered group, in general. Unlike in semigroups (without
order) if an ordered semigroup *S* is regular and has the
property *a S* ⊆ (*Sa]* for all *a ∈ S*, then the
*N*-classes of *S* are not left simple and right cancellative, in
general. The converse of the above two statements hold both in
semigroups and in ordered semigroups. Exactly as in semigroups
(without order), an ordered semigroup is a left group if and only
if it is regular and right cancellative.

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