View source for Periodic semi-group

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A [[Semi-group|semi-group]] in which each monogenic sub-semi-group (cf. [[Monogenic semi-group|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|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|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|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|Brandt semi-group]] $B_2$, which is isomorphic to a [[Rees semi-group of matrix type|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|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 [[#References|[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|Archimedean semi-group]]; 2) all idempotents in $S$ are pairwise incomparable with respect to the natural partial order (see [[Idempotent|Idempotent]]); and 3) $S$ is an ideal extension of a [[Completely-simple semi-group|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. [[#References|[5]]]); b) in $S$, each regular $\mathcal D$-class is a sub-semi-group; and c) each [[Regular element|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|nilpotent semi-group]] or a semi-group of idempotents (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]) [[#References|[4]]].

An important subclass of periodic semi-groups is constituted by the locally finite semi-groups (cf. [[Locally finite semi-group|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "The algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc.  (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.S. Prosvirov,   "Periodic semigroups"  ''Mat. Zap. Uralsk. Univ.'' , '''8''' :  1  (1971)  pp. 77–94  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Putcha,   "Semilattice decompositions of semigroups"  ''Semigroup Forum'' , '''6''' :  1  (1973)  pp. 12–34</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Schwarz,   "Contribution to the theory of torsion semigroups"  ''Chekhoslov. Mat. Zh.'' , '''3'''  (1953)  pp. 7–21  (In Russian)  (English abstract)</TD></TR></table>