CLASSES OF SEMIGROUPS
DEFINED VIA PROPERTIES OF REGULAR PARTS OF THEIR SUBSEMIGROUPS

Melanija Mitrovic

Date: University of Niš, Faculty of Mechanical Engineering,
Beogradska 14, 18000 Niš, Yugoslavia
e-mail: meli@junis.ni.ac.yu

For any semigroup $S$, we define

$$E(S)={e\in S\mid e^2=e},$$

$$Reg(S)=\{a\in S\mid (\exists\ x\in S)\ a=axa\},$$

$$LReg(S)=\{a\in S\mid (\exists\ x\in S)\ a=xa^2\},$$

$$Gr(S)=\{a\in S\mid (\exists\ x\in S)\ a=axa\ \ ax=xa\}.$$

For a subsemigroup $T$ of $S$, we define $reg(T)=T\cap Reg(S)$. Clearly,

$$Reg(T)\subseteq reg(T), \eqno{(1)}$$

and the inclusion can be strict (since every semigroup can be embedded into a regular one). For certain kinds of subsemigroups, (1) becomes an equality: for instance, this is the case when $T$ is a two-sided ideal of $S$ or a local submonoid $eSe$ where $e\in E(S)$. Imposing the restriction that (1) becomes an equality for certain other types of subsemigroups, we may distinguish interesting classes of semigroups. As an example, we present here

Theorem 1   The following conditions on a semigroup $S$ are equivalent: (i) $Gr(S)=Reg(S)$; (ii) $Reg(S)\subseteq LReg(S)$; (iii) $(\forall e\in E(S))\ reg(Se)=Reg(Se)$; (iv) $(\forall e,f\in E(S))\ reg(eSf)=Reg(eSf)$.

A natural question in this direction is the following: what are semigroups $S$ such that $Reg(T)=reg(T)$ for every subsemigroup $T$ of $S$. In order to answer this question, we need several notions. Recall that a semigroup $S$ is called completely $\pi$-regular (periodic) if for any element $a$ of $S$ there is a positive integer $k$ such that $a^k\in Gr(S)$ (respectively $a^k\in E(S)$). A semigroup $S$ is called archimedean if for every $a,b\in S$ there exists a positive integer $k$ such that $a^k\in SbS$. An archimedean and completely $\pi$-regular semigroup is said to be completely archimedean. A semilattice of completely archimedean semigroups is called a uniformly $\pi$-regular semigroup. If $\mathcal{K}$ is a class of semigroups, a hereditarily $\mathcal{K}$-semigroup is such that each of its subsemigroups belongs to $\mathcal K$. Finally, a semigroup is completely hereditarily archimedean if it is hereditarily archimedean and completely $\pi$-regular.

Theorem 2   The following condition on a semigroup $S$ are equivalent: (i) $reg(T)=Reg(T)\ne \emptyset$ for every subsemigroup $T$ of $S$; (ii) $S$ is a semilattice of completely hereditarily archimedean semigroups; (iii) $S$ is hereditarily uniformly $\pi$-regular semigroup; (iv) $S$ is periodic and $Reg(S)=Gr(S)$.