Difference between revisions of "Ordered semi-group"
m (link) |
(TeX partly done) |
||
Line 1: | Line 1: | ||
− | A [[ | + | 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 |
− | + | $$ | |
− | + | c(a \vee b) = ca \vee cb\ \ \text{and}\ \ (a \vee b)c = ac \vee bc | |
− | + | $$ | |
− | then | + | 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, | + | 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. | 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 | + | 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 $S$ 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.== | ==Totally ordered semi-groups.== | ||
− | A semi-group | + | 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. [[ | + | 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 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: | + | 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. |
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. | 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. | ||
Line 32: | Line 32: | ||
====References==== | ====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> | + | <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> | ||
+ | |||
+ | {{TEX|part}} |
Revision as of 16:39, 11 June 2016
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 (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-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 for all
). 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 of totally ordered semi-groups is called an
-homomorphism if
is an isotone mapping from
to
. A congruence (cf. Congruence (in algebra)) on a totally ordered semi-group is called an
-congruence if all its classes are convex subsets; the kernel congruences of
-homomorphisms are precisely the
-congruences. The decomposition of a totally ordered semi-group
into Archimedean classes does not always define
-congruences, i.e. they are not always a band (cf. Band of semi-groups), but this is so, for example, if
is periodic and its idempotents commute or if
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 -simple, semi-groups); a trivial example of such a totally ordered semi-group is a totally ordered group. A totally ordered semi-group
with a least element
and a greatest element
(in particular, finite) will be convex ideally simple if and only if
and
are at the same time left and right zeros in
. Any totally ordered semi-group may be imbedded, while preserving the order (
-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
-isomorphically imbedded in an Abelian totally ordered group; moreover, there exists a unique group of fractions, up to an
-isomorphism. A totally ordered semi-group is
-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
such that either
,
for all
, or
,
for all
).
Lattice-ordered semi-groups.
If for two elements and
in an ordered semi-group there exists a greatest element
with the property
, then it is called a right quotient and is denoted by
, the left quotient
is defined similarly. A lattice-ordered semi-group
is called a lattice-ordered semi-group with division if the right and left quotients exist in
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
,
. 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
. If the lattice-ordered semi-group
being considered is commutative, then the relation
is a congruence on it; moreover, the quotient semi-group
is a (lattice-ordered) group if and only if
is integrally closed, i.e.
for every
.
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 |
Ordered semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_semi-group&oldid=35779