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A [[semi-group]] equipped with a (generally speaking, partial) [[Order (on a set)|order]] $\le$ which is stable relative to the semi-group operation, i.e. for any elements $a,b,c$ it follows from $a \le b$ that $ac \le bc$ and $ca \le cb$. If the relation $\le$ on the ordered semi-group $S$ is a [[total order]], then $S$ is called a totally ordered semi-group (cf. also [[Totally ordered set]]). If the relation $\le$ on $S$ defines a [[lattice]] (with the associated operations join $\vee$ and meet $\wedge$) satisfying the identities
 
A [[semi-group]] equipped with a (generally speaking, partial) [[Order (on a set)|order]] $\le$ which is stable relative to the semi-group operation, i.e. for any elements $a,b,c$ it follows from $a \le b$ that $ac \le bc$ and $ca \le cb$. If the relation $\le$ on the ordered semi-group $S$ is a [[total order]], then $S$ is called a totally ordered semi-group (cf. also [[Totally ordered set]]). If the relation $\le$ on $S$ defines a [[lattice]] (with the associated operations join $\vee$ and meet $\wedge$) satisfying the identities
 
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A semi-group $S$ 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 $n$-step nilpotent semi-groups. There exists a continuum of methods for ordering free semi-groups of finite rank $\ge 2$. 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).
 
A semi-group $S$ 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 $n$-step nilpotent semi-groups. There exists a continuum of methods for ordering free semi-groups of finite rank $\ge 2$. 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]]) whose rectangular components are singular while the corresponding semi-lattices are trees. The completely-simple totally ordered semi-groups are exhausted by the [[right group]]s and the left groups and are lexicographic products (cf. [[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-group]]s, as well as a characterization in this way of the inverse totally ordered semi-groups (cf. [[Inversion semi-group]]). All types of totally ordered semi-groups generated by two mutually inverse elements have been classified (cf. [[Regular element]]).
+
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]]) whose rectangular components are singular while the corresponding semi-lattices are trees. The completely-simple totally ordered semi-groups are exhausted by the [[right group]]s and the left groups and are [[lexicographic product]]s 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-group]]s, as well as a characterization in this way of the inverse totally ordered semi-groups (cf. [[Inversion semi-group]]). All types of totally ordered semi-groups generated by two mutually inverse elements have been classified (cf. [[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-group]]s, 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.
+
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-group]]s, 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 $  x ^{2} \leq x $
 +
for all $  x $).  
 +
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.
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A [[Homomorphism|homomorphism]] $  \phi : \  A \rightarrow B $
 +
of totally ordered semi-groups is called an o $-
 +
homomorphism if $  \phi $
 +
is an [[Isotone mapping|isotone mapping]] from $  A $
 +
to $  B $.  
 +
A congruence (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) on a totally ordered semi-group is called an o $-
 +
congruence if all its classes are convex subsets; the kernel congruences of o $-
 +
homomorphisms are precisely the o $-
 +
congruences. The decomposition of a totally ordered semi-group $  S $
 +
into Archimedean classes does not always define o $-
 +
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 $  S $
 +
is periodic and its idempotents commute or if $  S $
 +
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  $  o $-
 +
simple, semi-groups); a trivial example of such a totally ordered semi-group is a totally ordered group. A totally ordered semi-group  $  S $
 +
with a least element  $  s $
 +
and a greatest element  $  g $(
 +
in particular, finite) will be convex ideally simple if and only if  $  s $
 +
and  $  g $
 +
are at the same time left and right zeros in  $  S $.  
 +
Any totally ordered semi-group may be imbedded, while preserving the order ( $  o $-
 +
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  $  o $-
 +
isomorphically imbedded in an Abelian totally ordered group; moreover, there exists a unique group of fractions, up to an  $  o $-
 +
isomorphism. A totally ordered semi-group is  $  o $-
 +
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  $  a ,\  b $
 +
such that either  $  a ^{n} < b ^{n+1} $,
 +
$  b ^{n} < a ^{n+1} $
 +
for all  $  n > 0 $,
 +
or  $  a ^{n} > b ^{n+1} $,
 +
$  b ^{n} > a ^{n+1} $
 +
for all  $  n > 0 $).
  
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.==
 
==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" />.
+
If for two elements $  a $
 +
and $  b $
 +
in an ordered semi-group there exists a greatest element $  x $
 +
with the property $  b x < a $,  
 +
then it is called a right quotient and is denoted by $  a : b $,  
 +
the left quotient $  a : b $
 +
is defined similarly. A lattice-ordered semi-group $  S $
 +
is called a lattice-ordered semi-group with division if the right and left quotients exist in $  S $
 +
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 $  a ( \lor _ \alpha  b _ \alpha  ) = \lor _ \alpha  ab _ \alpha  $,  
 +
$  ( \lor _ \alpha  b _ \alpha  ) a = \lor _ \alpha  b _ \alpha  a $.  
 +
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 $  a \sim b $
 +
$  \iff $
 +
$  1 : a = 1 : b $.  
 +
If the lattice-ordered semi-group $  S $
 +
being considered is commutative, then the relation $  \sim $
 +
is a congruence on it; moreover, the quotient semi-group $  S / \sim $
 +
is a (lattice-ordered) group if and only if $  S $
 +
is integrally closed, i.e. $  a : a = 1 $
 +
for every $  a \in S $.
 +
 
  
 
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 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.
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<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>
 
<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>
 
</table>
 
{{TEX|part}}
 

Latest revision as of 14:07, 17 March 2020


A semi-group equipped with a (generally speaking, partial) order $\le$ which is stable relative to the semi-group operation, i.e. for any elements $a,b,c$ it follows from $a \le b$ that $ac \le bc$ and $ca \le cb$. If the relation $\le$ on the ordered semi-group $S$ is a total order, then $S$ is called a totally ordered semi-group (cf. also Totally ordered set). If the relation $\le$ on $S$ defines a lattice (with the associated operations join $\vee$ and meet $\wedge$) satisfying the identities $$ c(a \vee b) = ca \vee cb\ \ \text{and}\ \ (a \vee b)c = ac \vee bc $$ then $S$ 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). On a lattice-ordered semi-group the identities $$ c(a \wedge b) = ca \wedge cb\ \ \text{and}\ \ (a \wedge b)c = ac \wedge bc $$ generally speaking, are not required to 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); their theory forms an independent part of algebra. In distinction to ordered groups, the order relation on an arbitrary ordered semi-group $S$ is, generally speaking, not defined by the set of its positive elements (i.e. the elements $a$ such that $ax \ge x$ and $xa \ge x$ for any $x$).

Totally ordered semi-groups.

A semi-group $S$ 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 $n$-step nilpotent semi-groups. There exists a continuum of methods for ordering free semi-groups of finite rank $\ge 2$. 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) 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 and the left groups and are lexicographic products 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, as well as a characterization in this way of the inverse totally ordered semi-groups (cf. Inversion semi-group). All types of totally ordered semi-groups generated by two mutually inverse elements have been classified (cf. 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, naturally totally ordered semi-groups (cf. Naturally ordered groupoid), positive ordered semi-groups (in which all elements are positive), integral totally ordered semi-groups (in which $ x ^{2} \leq x $ for all $ x $). 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). 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. 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 $ \phi : \ A \rightarrow B $ of totally ordered semi-groups is called an $ o $- homomorphism if $ \phi $ is an isotone mapping from $ A $ to $ B $. A congruence (cf. Congruence (in algebra)) on a totally ordered semi-group is called an $ o $- congruence if all its classes are convex subsets; the kernel congruences of $ o $- homomorphisms are precisely the $ o $- congruences. The decomposition of a totally ordered semi-group $ S $ into Archimedean classes does not always define $ o $- congruences, i.e. they are not always a band (cf. Band of semi-groups), but this is so, for example, if $ S $ is periodic and its idempotents commute or if $ S $ is a positive totally ordered semi-group.

For a totally ordered semi-group there arises an additional condition of simplicity (cf. Simple semi-group), related to the order. One such condition is the lack of proper convex ideals (convex ideally-simple, or $ o $- simple, semi-groups); a trivial example of such a totally ordered semi-group is a totally ordered group. A totally ordered semi-group $ S $ with a least element $ s $ and a greatest element $ g $( in particular, finite) will be convex ideally simple if and only if $ s $ and $ g $ are at the same time left and right zeros in $ S $. Any totally ordered semi-group may be imbedded, while preserving the order ( $ o $- 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 $ o $- isomorphically imbedded in an Abelian totally ordered group; moreover, there exists a unique group of fractions, up to an $ o $- isomorphism. A totally ordered semi-group is $ o $- 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 $ a ,\ b $ such that either $ a ^{n} < b ^{n+1} $, $ b ^{n} < a ^{n+1} $ for all $ n > 0 $, or $ a ^{n} > b ^{n+1} $, $ b ^{n} > a ^{n+1} $ for all $ n > 0 $).


Lattice-ordered semi-groups.

If for two elements $ a $ and $ b $ in an ordered semi-group there exists a greatest element $ x $ with the property $ b x < a $, then it is called a right quotient and is denoted by $ a : b $, the left quotient $ a : b $ is defined similarly. A lattice-ordered semi-group $ S $ is called a lattice-ordered semi-group with division if the right and left quotients exist in $ S $ 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 $ a ( \lor _ \alpha b _ \alpha ) = \lor _ \alpha ab _ \alpha $, $ ( \lor _ \alpha b _ \alpha ) a = \lor _ \alpha b _ \alpha a $. 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 $ a \sim b $ $ \iff $ $ 1 : a = 1 : b $. If the lattice-ordered semi-group $ S $ being considered is commutative, then the relation $ \sim $ is a congruence on it; moreover, the quotient semi-group $ S / \sim $ is a (lattice-ordered) group if and only if $ S $ is integrally closed, i.e. $ a : a = 1 $ for every $ a \in S $.


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) 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

[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[2] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[3] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[4] Itogi Nauk. Algebra. Topol. Geom. 1965 (1967) pp. 116–120
[5] Itogi Nauk. Algebra. Topol. Geom. 1966 (1968) pp. 99–102
[6] M. Satyanarayana, "Positively ordered semigroups" , M. Dekker (1979)
[7] 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
How to Cite This Entry:
Ordered semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_semi-group&oldid=38959
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article