A $$poe$$ -semigroup $$(S,\cdot,\leq)$$ equipped with an order preserving involution ‘‘ $$*$$ ’’ is called an involution ordered semigroup. An $$le$$ -semigroup is a lattice ordered semigroup possessing a greatest element. The concepts of $$*$$ -regularity, $$*$$ -intra-regularity, $$*$$ -bi-ideal element and $$*$$ -quasi-ideal element in this type of semigroups have been introduced and, using the right and left ideal elements, relationships among them are given. The $$*$$ -regular (resp. $$*$$ -intra-regular) $$poe$$ -semigroups are regular (resp. intra-regular). It is shown, among others, that in an involution $$*$$ -regular $$\vee e$$ -semigroup every $$*$$ -bi-ideal element can be represented as the product of a right and a left ideal element. Necessary and sufficient conditions are given under which an involution $$le$$ -semigroup is $$*$$ -regular, regular, $$*$$ -intra-regular or intra-regular. An involution $$poe$$ -semigroup $$S$$ is $$*$$ -intra-regular if and only if the ideal elements of $$S$$ are $$*$$ -semiprime; and an analogous result for involution $$*$$ -left or $$*$$ -right $$poe$$ -semigroups also holds. It is also shown that in an involution $$*$$ -intra-regular $$poe$$ -semigroup $$S$$ , the filter generated by an element $$x$$ of $$S$$ has a very simple form that is useful for the investigation. As a consequence, in this type of semigroups the element $$ex^{*}e$$ is contained in the $$(x^{*})_{\mathcal{N}}$$ -class (for any $$x$$ ) and it is the greatest element of the class $$(x)_{\mathcal{N}}$$ .