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Orthodox ordered semigroups
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DOI：10.1007/s00233-026-10613-x.png)
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Abstract
En 中文
An element e of an ordered semigroup (S, , <=) is called idempotent (resp. generalised idempotent) if e <= e(2) (resp. (e, e(2)) is an element of R-<= where R-<= is the smallest congruence on S containing the relation <= boolean OR <=(-1)). The set of all idempotents (resp. generalised idempotents) of S is denoted by E (S) (resp. E-G (S)). S is called orthodox if (i) the set E (S) is non empty and (ii) ef is an element of E-G (S) for every e, f is an element of E (S). An element x in S is an inverse (resp. generalised inverse) of an element a of S if a <= axa and x <= xax (resp. (a, axa) , (x, xax) is an element of R-<=). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a, then we know the set of all generalised inverses of a. We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.
Keywords:
Idempotent element of an ordered semigroup
Idempotent element of an ordered semigroup
Inverse element of an ordered semigroup
Generalised idempotent element of an ordered semigroup
Orthodox
Orthodoxordered semigroup
Relation R-<=
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