Periodic semi-group

A semi-group in which each monogenic sub-semi-group (cf. Monogenic semi-group) is finite (in other words, each element has finite order). Every periodic semi-group has idempotents. The set $K_e$ of all elements in a periodic semi-group some power (depending on the element) of which is equal to a given idempotent $e$ is called the torsion class corresponding to that idempotent. The set $G_e$ of all elements from $K_e$ for which $e$ serves as the unit is an $\mathcal H$-class (see Green equivalence relations). It is the largest subgroup in $K_e$ and an ideal in the sub-semi-group $\langle K_e\rangle$ generated by $K_e$; therefore, $\langle K_e\rangle$ is a homogroup (see Minimal ideal). A periodic semi-group containing a unique idempotent is called unipotent. The unipotency of a periodic semi-group $S$ is equivalent to each of the following conditions: $S$ is an ideal extension of a group by a nil semi-group, or $S$ is a subdirect product of a group and a nil semi-group.

The decomposition of a periodic semi-group into torsion classes plays a decisive part in the study of many aspects of periodic semi-groups. An arbitrary torsion class is not necessarily a sub-semi-group: A minimal counterexample is the five-element Brandt semi-group $B_2$, which is isomorphic to a Rees semi-group of matrix type over the unit group having as unit the sandwich matrix of order two. In a periodic semi-group $S$, all torsion classes are sub-semi-groups if and only if $S$ does not contain sub-semi-groups that are ideal extensions of a unipotent semi-group by $B_2$; in this case, the decomposition of $S$ into torsion classes is not necessarily a band of semi-groups. Various conditions are known (including necessary and sufficient ones) under which a periodic semi-group is a band of torsion classes; this clearly occurs for commutative semi-groups, and it is true for periodic semi-groups having two idempotents [3].

The Green relations $\mathcal D$ and $\mathcal J$ coincide in any periodic semi-group; a $0$-simple periodic semi-group is completely $0$-simple. The following conditions are equivalent for a periodic semi-group $S$: 1) $S$ is an Archimedean semi-group; 2) all idempotents in $S$ are pairwise incomparable with respect to the natural partial order (see Idempotent); and 3) $S$ is an ideal extension of a completely-simple semi-group by a nil semi-group. Many conditions equivalent to the fact that a periodic semi-group $S$ decomposes into a band (and then also into a semi-lattice) of Archimedean semi-groups are known; they include the following: a) for any $a\in S$ and for any idempotent $e\in S$, if $e\in SaS$, then $e\in Sa^2S$ (cf. [5]); b) in $S$, each regular $\mathcal D$-class is a sub-semi-group; and c) each regular element of $S$ is a group element.

Let $S$ be an infinite periodic semi-group and let $E_S$ be the set of all its idempotents. If $E_S$ is finite, $S$ contains an infinite unipotent sub-semi-group, while if $E_S$ is infinite, $S$ contains an infinite sub-semi-group that is a nilpotent semi-group or a semi-group of idempotents (cf. Idempotents, semi-group of) [4].

An important subclass of periodic semi-groups is constituted by the locally finite semi-groups (cf. Locally finite semi-group). A more extensive class is constituted by the quasi-periodic semi-groups ($S$ is called quasi-periodic if some power of each of its elements lies in a subgroup $G\subseteq S$). Many properties of periodic semi-groups can be transferred to quasi-periodic ones. Quasi-periodic groups are also called epigroups.

References

[1]	A.H. Clifford, G.B. Preston, "The algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961)
[2]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[3]	A.S. Prosvirov, "Periodic semigroups" Mat. Zap. Uralsk. Univ. , 8 : 1 (1971) pp. 77–94 (In Russian)
[4]	L.N. Shevrin, "On the theory of periodic semigroups" Soviet Math. Izv. Vyz. , 18 : 5 (1974) pp. 172–181 Izv. Vyzov. Mat. , 18 : 5 (1974) pp. 205–215
[5]	M. Putcha, "Semilattice decompositions of semigroups" Semigroup Forum , 6 : 1 (1973) pp. 12–34
[6]	S. Schwarz, "Contribution to the theory of torsion semigroups" Chekhoslov. Mat. Zh. , 3 (1953) pp. 7–21 (In Russian) (English abstract)