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 of all elements in a periodic semi-group some power (depending on the element) of which is equal to a given idempotent is called the torsion class corresponding to that idempotent. The set of all elements from for which serves as the unit is an -class (see Green equivalence relations). It is the largest subgroup in and an ideal in the sub-semi-group generated by ; therefore, is a homogroup (see Minimal ideal). A periodic semi-group containing a unique idempotent is called unipotent. The unipotency of a periodic semi-group is equivalent to each of the following conditions: is an ideal extension of a group by a nil semi-group, or 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 , 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 , all torsion classes are sub-semi-groups if and only if does not contain sub-semi-groups that are ideal extensions of a unipotent semi-group by ; in this case, the decomposition of 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 and coincide in any periodic semi-group; a -simple periodic semi-group is completely -simple. The following conditions are equivalent for a periodic semi-group : 1) is an Archimedean semi-group; 2) all idempotents in are pairwise incomparable with respect to the natural partial order (see Idempotent); and 3) 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 decomposes into a band (and then also into a semi-lattice) of Archimedean semi-groups are known; they include the following: a) for any and for any idempotent , if , then (cf. [5]); b) in , each regular -class is a sub-semi-group; and c) each regular element of is a group element.

Let be an infinite periodic semi-group and let be the set of all its idempotents. If is finite, contains an infinite unipotent sub-semi-group, while if is infinite, 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 ( is called quasi-periodic if some power of each of its elements lies in a subgroup ). 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)