A semigroup S is called eventually regular (or π-regular) if for every element a of S there exists m ∈ Z~+ (the set of positive integers) such that a~m is regular. Let us denote by r(a) the least positive integer m such that a~m is regular of S and call it the regular index of an element a. If every regular element of a π-regular semigroup S possesses a unique inverse, then S is called eventually inverse (π-inverse) [1], [12]. Let A be a subsemigroup of an eventually inverse semigroup S. We say that A is an eventually inverse subsemigroup of S if for any a ∈ RegA (the set of all regular elements of A) [9]. Obviously, A is an eventually inverse subsemigroup of S if and only if for any a ∈ A and every m ∈ Z~+, a~m RegS implies a~m ∈RegA. For an eventually inverse semigroup S, the set SubπS of all eventually inverse subsemigroups (including the empty set) of S forms a lattice with respect to intersection denoted as usual by ∩ and union denoted by π<,>, where π is the eventually inverse subsemigroup generated by the union of subsets A, B of S.
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