# Regular element

of a semi-group

An element $a$ such that $a=axa$ for some element $x$ of the given semi-group; if in addition (for the same $x$) $ax=xa$, then $a$ is called completely regular. If $a$ is a regular element of a semi-group $S$, then the principal right (left) ideal in $S$ generated by $a$ is generated by some idempotent; conversely, each of these symmetrical properties implies the regularity of $a$. If $aba=a$ and $bab=b$, the elements $a$ and $b$ are called mutually inverse (also known as generalized inverse or regularly conjugate). Every regular element has an element inverse to it; generally speaking, it is not unique (see Inversion semi-group). Semi-groups in which any two elements are mutually inverse are in fact rectangular semi-groups (see Idempotents, semi-group of). Each completely-regular element $a$ has an element inverse to it that commutes with $a$. An element is completely regular if and only if it belongs to some subgroup of a semi-group (cf. Clifford semi-group). For regular $\mathcal D$-classes, see Green equivalence relations.

#### References

 [1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) [2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)