View source for Ordered semi-group

A [[Semi-group|semi-group]] equipped with a (generally speaking, partial) order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701501.png" /> which is stable relative to the semi-group operation, i.e. for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701502.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701503.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701505.png" />. If the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701506.png" /> on the ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701507.png" /> is a total order, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701508.png" /> is called a totally ordered semi-group (cf. also [[Totally ordered set|Totally ordered set]]). If the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o0701509.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015010.png" /> defines a [[Lattice|lattice]] (with the associated operations union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015011.png" /> and intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015012.png" />) satisfying the identities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015013.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015014.png" /> is called a lattice-ordered semi-group; thus, the class of all lattice-ordered semi-groups, considered as algebras with semi-group and lattice operations, is a variety (cf. also [[Variety of groups|Variety of groups]]). On a lattice-ordered semi-group the identities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015015.png" /></td> </tr></table>

generally speaking, do not hold, and their imposition singles out a proper subvariety of the variety of all lattice-ordered semi-groups.

Ordered semi-groups arise by considering different numerical semi-groups, semi-groups of functions and binary relations, semi-groups of subsets (or subsystems of different algebraic systems, for example ideals in rings and semi-groups), etc. Every ordered semi-group is isomorphic to a certain semi-group of binary relations, considered as an ordered semi-group, where the order is set-theoretic inclusion. The classical example of a lattice-ordered semi-group is the semi-group of all binary relations on an arbitrary set.

In the general theory of ordered semi-groups one can distinguish two main developments: the theory of totally ordered semi-groups and the theory of lattice-ordered semi-groups. Although every totally ordered semi-group is lattice-ordered, both theories have developed to a large degree independently. The study of totally ordered semi-groups is devoted to properties that are to a large extent not shared by lattice-ordered semi-groups, while in considering lattice-ordered semi-groups one studies as a rule properties which, when applied to totally ordered semi-groups, reduce to degenerate cases. An important type of semi-groups is formed by the ordered groups (cf. [[Ordered group|Ordered group]]); their theory forms an independent part of algebra. In distinction to ordered groups, the order relation on an arbitrary ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015016.png" /> is, generally speaking, not defined by the set of its positive elements (i.e. the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015019.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015020.png" />).

==Totally ordered semi-groups.==
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015021.png" /> is called orderable if one can define on it a total order which turns it into a totally ordered semi-group. A necessary condition for orderability is the absence in the semi-group of non-idempotent elements of finite order. If in an orderable semi-group the set of all idempotents is non-empty, then it is a sub-semi-group. Among the orderable semi-groups are the free semi-groups, the free commutative semi-groups, and the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015022.png" />-step nilpotent semi-groups. There exists a continuum of methods for ordering free semi-groups of finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015023.png" />. Certain necessary and sufficient conditions for the orderability of arbitrary semi-groups have been found, as well as for semi-groups from a series of known classes (e.g. semi-groups of idempotents, inverse semi-groups).

The structure of totally ordered semi-groups of idempotents has been completely described; in particular, the decomposition of such semi-groups into semi-lattices of rectangular semi-groups (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]) whose rectangular components are singular while the corresponding semi-lattices are trees. The completely-simple totally ordered semi-groups are exhausted by the right groups (cf. [[Right group|Right group]]) and the left groups and are lexicographic products (cf. [[Lexicographic order|Lexicographic order]]) of totally ordered groups and totally ordered semi-groups of right (respectively, left) zeros. By applying the reduction to totally ordered groups, a description of totally ordered semi-groups has been obtained in terms of the class of Clifford semi-groups (cf. [[Clifford semi-group|Clifford semi-group]]), as well as a characterization in this way of the inverse totally ordered semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]). All types of totally ordered semi-groups generated by two mutually inverse elements have been classified (cf. [[Regular element|Regular element]]).

The conditions imposed in the study of totally ordered semi-groups often postulate additional connections between the operation and the order relation. In this way one distinguishes the following basic types of totally ordered semi-groups: Archimedean semi-groups (cf. [[Archimedean semi-group|Archimedean semi-group]]), naturally totally ordered semi-groups (cf. [[Naturally ordered groupoid|Naturally ordered groupoid]]), positive ordered semi-groups (in which all elements are positive), integral totally ordered semi-groups (in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015025.png" />). Every Archimedean naturally totally ordered semi-group is commutative; their structure is completely described. The structure of an arbitrary totally ordered semi-group is to a large extent determined by the peculiarities of its decomposition into Archimedean classes (cf. [[Archimedean class|Archimedean class]]). For a periodic totally ordered semi-group this decomposition coincides with the decomposition into torsion classes, and, moreover, each Archimedean class is a [[Nil semi-group|nil semi-group]]. An arbitrary totally ordered nil semi-group is the union of an increasing sequence of convex nilpotent sub-semi-groups; in particular, it is locally nilpotent.

A [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015026.png" /> of totally ordered semi-groups is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015028.png" />-homomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015029.png" /> is an [[Isotone mapping|isotone mapping]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015030.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015031.png" />. A congruence (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) on a totally ordered semi-group is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015033.png" />-congruence if all its classes are convex subsets; the kernel congruences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015034.png" />-homomorphisms are precisely the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015035.png" />-congruences. The decomposition of a totally ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015036.png" /> into Archimedean classes does not always define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015037.png" />-congruences, i.e. they are not always a band (cf. [[Band of semi-groups|Band of semi-groups]]), but this is so, for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015038.png" /> is periodic and its idempotents commute or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015039.png" /> is a positive totally ordered semi-group.

For a totally ordered semi-group there arises an additional condition of simplicity (cf. [[Simple semi-group|Simple semi-group]]), related to the order. One such condition is the lack of proper convex ideals (convex ideally-simple, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015041.png" />-simple, semi-groups); a trivial example of such a totally ordered semi-group is a totally ordered group. A totally ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015043.png" /> with a least element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015044.png" /> and a greatest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015045.png" /> (in particular, finite) will be convex ideally simple if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015047.png" /> are at the same time left and right zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015048.png" />. Any totally ordered semi-group may be imbedded, while preserving the order (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015049.png" />-isomorphically), in a convex ideally-simple totally ordered semi-group. There exist totally ordered semi-groups with cancellation, non-imbeddable in a group, but a commutative totally ordered semi-group with cancellation can be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015050.png" />-isomorphically imbedded in an Abelian totally ordered group; moreover, there exists a unique group of fractions, up to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015051.png" />-isomorphism. A totally ordered semi-group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015052.png" />-isomorphically imbeddable in the additive group of real numbers if and only if it satisfies the [[cancellation law]] and contains no abnormal pair (i.e. elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015053.png" /> such that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015055.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015056.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015058.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015059.png" />).

==Lattice-ordered semi-groups.==
If for two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015061.png" /> in an ordered semi-group there exists a greatest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015062.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015063.png" />, then it is called a right quotient and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015064.png" />, the left quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015065.png" /> is defined similarly. A lattice-ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015066.png" /> is called a lattice-ordered semi-group with division if the right and left quotients exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015067.png" /> for any pair of elements. Such semi-groups are complete (as a lattice) lattice-ordered semi-groups, their lattice zero is also the multiplicative zero and they satisfy the infinite distributive laws <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015069.png" />. An important example of a lattice-ordered semi-group with division is the multiplicative semi-group of ideals of an associative ring, and a notable direction in the theory of lattice-ordered semi-groups deals with the transfer of many properties and results from the theory of ideals in associative rings to the case of lattice-ordered semi-groups (the unique decomposition into prime factors, primes, primary, maximal, principal elements of a lattice-ordered semi-group, etc.). For example, the well-known relation of Artin in the theory of commutative rings can be translated as follows in the theory of lattice-ordered semi-groups with division and having a one 1: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015070.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015071.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015072.png" />. If the lattice-ordered semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015073.png" /> being considered is commutative, then the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015074.png" /> is a congruence on it; moreover, the quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015075.png" /> is a (lattice-ordered) group if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015076.png" /> is integrally closed, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015077.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070150/o07015078.png" />.

The study of lattice-ordered semi-groups is connected with groups by considering the imbedding problems of a lattice-ordered semi-group in a lattice-ordered group. For example, every lattice-ordered semi-group with cancellation and the Ore condition (cf. [[Imbedding of semi-groups|Imbedding of semi-groups]]) and whose multiplication is distributive relative to both lattice operations, is imbeddable in a lattice-ordered group.

The theory of lattice-ordered semi-groups has begun to be studied from the point of view of the theory of varieties: the free lattice-ordered semi-groups have been described, the minimal varieties of lattice-ordered semi-groups have been found, etc.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kokorin,   V.M. Kopytov,   "Fully ordered groups" , Israel Program Sci. Transl.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  ''Itogi Nauk. Algebra. Topol. Geom. 1965''  (1967)  pp. 116–120</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  ''Itogi Nauk. Algebra. Topol. Geom. 1966''  (1968)  pp. 99–102</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Satyanarayana,   "Positively ordered semigroups" , M. Dekker  (1979)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.Ya. Gabovich,   "Fully ordered semigroups and their applications"  ''Russian Math. Surveys'' , '''31''' :  1  (1976)  pp. 147–216  ''Uspekhi Mat. Nauk'' , '''31''' :  1  (1976)  pp. 137–201</TD></TR></table>