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A class of semi-groups (cf. [[Semi-group|Semi-group]]) defined by a system of identities, or laws (see [[Algebraic systems, variety of|Algebraic systems, variety of]]). Every variety of semi-groups is either periodic, i.e. it consists of periodic semi-groups, or overcommutative, i.e. it contains the variety of all commutative semi-groups. Various properties of varieties of semi-groups are classified by singling out certain types of identities. An identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963101.png" /> is said to be normal (also homotypical, regular or uniform) if the sets of variables figuring in the words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963103.png" /> are the same, and anomalous (or heterotypical) otherwise. An identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963104.png" /> is said to be balanced if each variable appears in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963105.png" /> just as many times as it does in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963106.png" />. A special case of a balanced identity is a permutation identity — if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963108.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v0963109.png" /> by permuting the variables. A variety of semi-groups is overcommutative if and only if all its identities are balanced. A basis of identities for a variety of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631010.png" /> is said to be irreducible if any of its proper subsets defines a variety distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631011.png" />. Every overcommutative variety of semi-groups has an irreducible basis of identities. There exist varieties of semi-groups which do not have irreducible bases of identities. Examples of varieties of semi-groups with finite bases are: any variety of commutative semi-groups; any periodic variety of semi-groups with a permutation identity; any variety of semi-groups defined by permutation identities. Any semi-group with less than six elements has a finite basis of identities, but there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631012.png" />-element semi-group that has no finite basis of identities.
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The following conditions for a variety of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631013.png" /> are equivalent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631014.png" /> is defined by normal identities; all identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631015.png" /> are normal; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631016.png" /> contains a two-element [[Semi-lattice|semi-lattice]]. Among the identities of a variety of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631017.png" /> there is an anomalous one if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631018.png" /> is periodic and consists of Archimedean semi-groups (cf. [[Archimedean semi-group|Archimedean semi-group]]).
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The minimal varieties of semi-groups are exhausted by the varieties of all: 1) semi-lattices; 2) semi-groups of left zeros; 3) semi-groups of right zeros (see [[Idempotents, semi-group of|Idempotents, semi-group of]]); 4) semi-groups with zero multiplication; 5) Abelian groups of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631019.png" /> for any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631020.png" />. In the lattice of all varieties of semi-groups, every non-unit element has an element that covers it; a unit element cannot be equal to the union of finitely many non-unit elements. The lattice of all varieties of semi-groups does not satisfy any non-trivial lattice identity and has the cardinality of the continuum. The sublattice of all varieties of nil-semi-groups with the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631021.png" /> is also of the cardinality of the continuum, as is the sublattice of all overcommutative varieties. For some varieties of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631022.png" />, explicit descriptions have been discovered for the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631023.png" /> of subvarieties of it; there are also descriptions of varieties of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631024.png" /> with certain restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631025.png" />.
+
A class of semi-groups (cf. [[Semi-group|Semi-group]]) defined by a system of identities, or laws (see [[Algebraic systems, variety of|Algebraic systems, variety of]]). Every variety of semi-groups is either periodic, i.e. it consists of periodic semi-groups, or overcommutative, i.e. it contains the variety of all commutative semi-groups. Various properties of varieties of semi-groups are classified by singling out certain types of identities. An identity $  u = v $
 +
is said to be normal (also homotypical, regular or uniform) if the sets of variables figuring in the words  $  u $
 +
and  $  v $
 +
are the same, and anomalous (or heterotypical) otherwise. An identity  $  u = v $
 +
is said to be balanced if each variable appears in  $  u $
 +
just as many times as it does in  $  v $.
 +
A special case of a balanced identity is a permutation identity — if  $  u = x _ {1} \dots x _ {m} $
 +
and  $  v $
 +
is obtained from  $  u $
 +
by permuting the variables. A variety of semi-groups is overcommutative if and only if all its identities are balanced. A basis of identities for a variety of semi-groups  $  \mathfrak M $
 +
is said to be irreducible if any of its proper subsets defines a variety distinct from  $  \mathfrak M $.
 +
Every overcommutative variety of semi-groups has an irreducible basis of identities. There exist varieties of semi-groups which do not have irreducible bases of identities. Examples of varieties of semi-groups with finite bases are: any variety of commutative semi-groups; any periodic variety of semi-groups with a permutation identity; any variety of semi-groups defined by permutation identities. Any semi-group with less than six elements has a finite basis of identities, but there exists a  $  6 $-
 +
element semi-group that has no finite basis of identities.
  
A variety of semi-groups is said to be small if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631026.png" /> is finite. A variety of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631027.png" /> is called a variety of finite index if the degrees of nilpotency of the nilpotent semi-groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631028.png" /> are uniformly bounded (equivalent conditions are: every nil-semi-group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631029.png" /> is nilpotent; or: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631030.png" /> does not contain the variety of all commutative nil-semi-groups with the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631031.png" />). Every small variety of semi-groups is of finite index.
+
The following conditions for a variety of semi-groups $  \mathfrak M $
 +
are equivalent: $  \mathfrak M $
 +
is defined by normal identities; all identities of  $  \mathfrak M $
 +
are normal;  $  \mathfrak M $
 +
contains a two-element [[Semi-lattice|semi-lattice]]. Among the identities of a variety of semi-groups $  \mathfrak M $
 +
there is an anomalous one if and only if $  \mathfrak M $
 +
is periodic and consists of Archimedean semi-groups (cf. [[Archimedean semi-group|Archimedean semi-group]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631032.png" /> is a periodic variety of semi-groups, the following conditions are equivalent [[#References|[4]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631033.png" /> consists of bands of Archimedean semi-groups; in any semi-group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631034.png" />, every torsion class is a sub-semi-group; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631035.png" /> does not contain the Brandt semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631036.png" /> (see [[Periodic semi-group|Periodic semi-group]]). These conditions are satisfied by varieties of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631037.png" /> with a modular lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631038.png" /> and varieties of semi-groups of finite index (in particular, small varieties). A small variety of semi-groups is locally finite (i.e. consists of locally finite semi-groups) if and only if the variety of all groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631039.png" /> is locally finite; the small locally finite varieties of groups are precisely the Cross varieties (see [[Variety of groups|Variety of groups]]). For other locally finite varieties of semi-groups, see [[Locally finite semi-group|Locally finite semi-group]]. There is a description of the varieties of semi-groups whose elements are residually-finite semi-groups [[#References|[3]]].
+
The minimal varieties of semi-groups are exhausted by the varieties of all: 1) semi-lattices; 2) semi-groups of left zeros; 3) semi-groups of right zeros (see [[Idempotents, semi-group of|Idempotents, semi-group of]]); 4) semi-groups with zero multiplication; 5) Abelian groups of exponent  $  p $
 +
for any prime number  $  p $.  
 +
In the lattice of all varieties of semi-groups, every non-unit element has an element that covers it; a unit element cannot be equal to the union of finitely many non-unit elements. The lattice of all varieties of semi-groups does not satisfy any non-trivial lattice identity and has the cardinality of the continuum. The sublattice of all varieties of nil-semi-groups with the identity  $  x  ^ {2} = 0 $
 +
is also of the cardinality of the continuum, as is the sublattice of all overcommutative varieties. For some varieties of semi-groups $  \mathfrak M $,  
 +
explicit descriptions have been discovered for the lattice  $  L \mathfrak M $
 +
of subvarieties of it; there are also descriptions of varieties of semi-groups $  \mathfrak M $
 +
with certain restrictions on  $  L \mathfrak M $.
  
The set of all varieties of semi-groups forms a partial groupoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631040.png" /> relative to the [[Mal'tsev product|Mal'tsev product]]. The idempotents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631041.png" /> are known; there are just nine of them. The set of all varieties of semi-groups defined by systems of identities of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631042.png" /> is a maximal [[Groupoid|groupoid]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096310/v09631043.png" />.
+
A variety of semi-groups is said to be small if  $  L \mathfrak M $
 +
is finite. A variety of semi-groups  $  \mathfrak M $
 +
is called a variety of finite index if the degrees of nilpotency of the nilpotent semi-groups in  $  \mathfrak M $
 +
are uniformly bounded (equivalent conditions are: every nil-semi-group in  $  \mathfrak M $
 +
is nilpotent; or:  $  \mathfrak M $
 +
does not contain the variety of all commutative nil-semi-groups with the identity  $  x  ^ {2} = 0 $).
 +
Every small variety of semi-groups is of finite index.
 +
 
 +
If  $  \mathfrak M $
 +
is a periodic variety of semi-groups, the following conditions are equivalent [[#References|[4]]]:  $  \mathfrak M $
 +
consists of bands of Archimedean semi-groups; in any semi-group in  $  \mathfrak M $,
 +
every torsion class is a sub-semi-group;  $  \mathfrak M $
 +
does not contain the Brandt semi-group  $  B _ {2} $(
 +
see [[Periodic semi-group|Periodic semi-group]]). These conditions are satisfied by varieties of semi-groups  $  \mathfrak M $
 +
with a modular lattice  $  L \mathfrak M $
 +
and varieties of semi-groups of finite index (in particular, small varieties). A small variety of semi-groups is locally finite (i.e. consists of locally finite semi-groups) if and only if the variety of all groups in  $  \mathfrak M $
 +
is locally finite; the small locally finite varieties of groups are precisely the Cross varieties (see [[Variety of groups|Variety of groups]]). For other locally finite varieties of semi-groups, see [[Locally finite semi-group|Locally finite semi-group]]. There is a description of the varieties of semi-groups whose elements are residually-finite semi-groups [[#References|[3]]].
 +
 
 +
The set of all varieties of semi-groups forms a partial groupoid $  G $
 +
relative to the [[Mal'tsev product|Mal'tsev product]]. The idempotents of $  G $
 +
are known; there are just nine of them. The set of all varieties of semi-groups defined by systems of identities of the type $  w = 0 $
 +
is a maximal [[Groupoid|groupoid]] in $  G $.
  
 
Studies have also been conducted on varieties of semi-groups with additional signature operations: varieties of monoids (with an identity, cf. [[Monoid|Monoid]]); varieties of semi-groups with a zero; varieties of inverse semi-groups; etc.
 
Studies have also been conducted on varieties of semi-groups with additional signature operations: varieties of monoids (with an identity, cf. [[Monoid|Monoid]]); varieties of semi-groups with a zero; varieties of inverse semi-groups; etc.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Evans,  "The lattice of semigroup varieties"  ''Semigroup Forum'' , '''2''' :  1  (1971)  pp. 1–43</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya Aizenshtat,  B.K. Boguta,  , ''Semi-group varieties and semi-groups of endomorphisms'' , Leningrad  (1979)  pp. 3–46  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Golubov,  M.V. Sapir,  "Varieties of finitely approximable semigroups"  ''Soviet Math. Dokl.'' , '''20''' :  4  (1979)  pp. 828–832  ''Dokl. Akad. Nauk SSSR'' , '''247''' :  5  (1979)  pp. 1037–1041</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.V. Sapir,  E.V. Sukhanov,  "On manifolds of periodic semigroups"  ''Soviet Math. Izv. Vyz.'' , '''25''' :  4  (1981)  pp. 53–63  ''Izv. Vuzov. Mat.'' , '''25''' :  4  (1981)  pp. 48–55</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.N. Shevrin,  M.V. Volkov,  "Identities of semigroups"  ''Soviet Math. Izv. Vyz.'' , '''29''' :  11  (1985)  pp. 1–64  ''Izv. Vuzov. Mat.'' , '''29''' :  11  (1985)  pp. 3–47</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.N. Shevrin,  E.V. Sukhanov,  "Structural aspects of theory of semigroup varieties"  ''Soviet Math. Izv. Vyz.'' , '''33''' :  6  (1989)  pp. 1–34  ''Izv. Vuzov. Mat.'' , '''33''' :  6  (1989)  pp. 3–39</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Evans,  "The lattice of semigroup varieties"  ''Semigroup Forum'' , '''2''' :  1  (1971)  pp. 1–43</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya Aizenshtat,  B.K. Boguta,  , ''Semi-group varieties and semi-groups of endomorphisms'' , Leningrad  (1979)  pp. 3–46  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Golubov,  M.V. Sapir,  "Varieties of finitely approximable semigroups"  ''Soviet Math. Dokl.'' , '''20''' :  4  (1979)  pp. 828–832  ''Dokl. Akad. Nauk SSSR'' , '''247''' :  5  (1979)  pp. 1037–1041</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.V. Sapir,  E.V. Sukhanov,  "On manifolds of periodic semigroups"  ''Soviet Math. Izv. Vyz.'' , '''25''' :  4  (1981)  pp. 53–63  ''Izv. Vuzov. Mat.'' , '''25''' :  4  (1981)  pp. 48–55</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.N. Shevrin,  M.V. Volkov,  "Identities of semigroups"  ''Soviet Math. Izv. Vyz.'' , '''29''' :  11  (1985)  pp. 1–64  ''Izv. Vuzov. Mat.'' , '''29''' :  11  (1985)  pp. 3–47</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.N. Shevrin,  E.V. Sukhanov,  "Structural aspects of theory of semigroup varieties"  ''Soviet Math. Izv. Vyz.'' , '''33''' :  6  (1989)  pp. 1–34  ''Izv. Vuzov. Mat.'' , '''33''' :  6  (1989)  pp. 3–39</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.S. [E.S. Lyapin] Ljapin,  "Semigroups" , Amer. Math. Soc.  (1978)  pp. Chapt. XII  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.R. Reilly,  "On the lattice of varieties of completely regular semigroups"  S.M. Goberstein (ed.)  P.M. Higgins (ed.) , ''Semigroups and Their Applications'' , Reidel  (1987)  pp. 153–167</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.S. [E.S. Lyapin] Ljapin,  "Semigroups" , Amer. Math. Soc.  (1978)  pp. Chapt. XII  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.R. Reilly,  "On the lattice of varieties of completely regular semigroups"  S.M. Goberstein (ed.)  P.M. Higgins (ed.) , ''Semigroups and Their Applications'' , Reidel  (1987)  pp. 153–167</TD></TR></table>

Revision as of 08:28, 6 June 2020


A class of semi-groups (cf. Semi-group) defined by a system of identities, or laws (see Algebraic systems, variety of). Every variety of semi-groups is either periodic, i.e. it consists of periodic semi-groups, or overcommutative, i.e. it contains the variety of all commutative semi-groups. Various properties of varieties of semi-groups are classified by singling out certain types of identities. An identity $ u = v $ is said to be normal (also homotypical, regular or uniform) if the sets of variables figuring in the words $ u $ and $ v $ are the same, and anomalous (or heterotypical) otherwise. An identity $ u = v $ is said to be balanced if each variable appears in $ u $ just as many times as it does in $ v $. A special case of a balanced identity is a permutation identity — if $ u = x _ {1} \dots x _ {m} $ and $ v $ is obtained from $ u $ by permuting the variables. A variety of semi-groups is overcommutative if and only if all its identities are balanced. A basis of identities for a variety of semi-groups $ \mathfrak M $ is said to be irreducible if any of its proper subsets defines a variety distinct from $ \mathfrak M $. Every overcommutative variety of semi-groups has an irreducible basis of identities. There exist varieties of semi-groups which do not have irreducible bases of identities. Examples of varieties of semi-groups with finite bases are: any variety of commutative semi-groups; any periodic variety of semi-groups with a permutation identity; any variety of semi-groups defined by permutation identities. Any semi-group with less than six elements has a finite basis of identities, but there exists a $ 6 $- element semi-group that has no finite basis of identities.

The following conditions for a variety of semi-groups $ \mathfrak M $ are equivalent: $ \mathfrak M $ is defined by normal identities; all identities of $ \mathfrak M $ are normal; $ \mathfrak M $ contains a two-element semi-lattice. Among the identities of a variety of semi-groups $ \mathfrak M $ there is an anomalous one if and only if $ \mathfrak M $ is periodic and consists of Archimedean semi-groups (cf. Archimedean semi-group).

The minimal varieties of semi-groups are exhausted by the varieties of all: 1) semi-lattices; 2) semi-groups of left zeros; 3) semi-groups of right zeros (see Idempotents, semi-group of); 4) semi-groups with zero multiplication; 5) Abelian groups of exponent $ p $ for any prime number $ p $. In the lattice of all varieties of semi-groups, every non-unit element has an element that covers it; a unit element cannot be equal to the union of finitely many non-unit elements. The lattice of all varieties of semi-groups does not satisfy any non-trivial lattice identity and has the cardinality of the continuum. The sublattice of all varieties of nil-semi-groups with the identity $ x ^ {2} = 0 $ is also of the cardinality of the continuum, as is the sublattice of all overcommutative varieties. For some varieties of semi-groups $ \mathfrak M $, explicit descriptions have been discovered for the lattice $ L \mathfrak M $ of subvarieties of it; there are also descriptions of varieties of semi-groups $ \mathfrak M $ with certain restrictions on $ L \mathfrak M $.

A variety of semi-groups is said to be small if $ L \mathfrak M $ is finite. A variety of semi-groups $ \mathfrak M $ is called a variety of finite index if the degrees of nilpotency of the nilpotent semi-groups in $ \mathfrak M $ are uniformly bounded (equivalent conditions are: every nil-semi-group in $ \mathfrak M $ is nilpotent; or: $ \mathfrak M $ does not contain the variety of all commutative nil-semi-groups with the identity $ x ^ {2} = 0 $). Every small variety of semi-groups is of finite index.

If $ \mathfrak M $ is a periodic variety of semi-groups, the following conditions are equivalent [4]: $ \mathfrak M $ consists of bands of Archimedean semi-groups; in any semi-group in $ \mathfrak M $, every torsion class is a sub-semi-group; $ \mathfrak M $ does not contain the Brandt semi-group $ B _ {2} $( see Periodic semi-group). These conditions are satisfied by varieties of semi-groups $ \mathfrak M $ with a modular lattice $ L \mathfrak M $ and varieties of semi-groups of finite index (in particular, small varieties). A small variety of semi-groups is locally finite (i.e. consists of locally finite semi-groups) if and only if the variety of all groups in $ \mathfrak M $ is locally finite; the small locally finite varieties of groups are precisely the Cross varieties (see Variety of groups). For other locally finite varieties of semi-groups, see Locally finite semi-group. There is a description of the varieties of semi-groups whose elements are residually-finite semi-groups [3].

The set of all varieties of semi-groups forms a partial groupoid $ G $ relative to the Mal'tsev product. The idempotents of $ G $ are known; there are just nine of them. The set of all varieties of semi-groups defined by systems of identities of the type $ w = 0 $ is a maximal groupoid in $ G $.

Studies have also been conducted on varieties of semi-groups with additional signature operations: varieties of monoids (with an identity, cf. Monoid); varieties of semi-groups with a zero; varieties of inverse semi-groups; etc.

References

[1] T. Evans, "The lattice of semigroup varieties" Semigroup Forum , 2 : 1 (1971) pp. 1–43
[2] A.Ya Aizenshtat, B.K. Boguta, , Semi-group varieties and semi-groups of endomorphisms , Leningrad (1979) pp. 3–46 (In Russian)
[3] E.A. Golubov, M.V. Sapir, "Varieties of finitely approximable semigroups" Soviet Math. Dokl. , 20 : 4 (1979) pp. 828–832 Dokl. Akad. Nauk SSSR , 247 : 5 (1979) pp. 1037–1041
[4] M.V. Sapir, E.V. Sukhanov, "On manifolds of periodic semigroups" Soviet Math. Izv. Vyz. , 25 : 4 (1981) pp. 53–63 Izv. Vuzov. Mat. , 25 : 4 (1981) pp. 48–55
[5] L.N. Shevrin, M.V. Volkov, "Identities of semigroups" Soviet Math. Izv. Vyz. , 29 : 11 (1985) pp. 1–64 Izv. Vuzov. Mat. , 29 : 11 (1985) pp. 3–47
[6] L.N. Shevrin, E.V. Sukhanov, "Structural aspects of theory of semigroup varieties" Soviet Math. Izv. Vyz. , 33 : 6 (1989) pp. 1–34 Izv. Vuzov. Mat. , 33 : 6 (1989) pp. 3–39

Comments

References

[a1] E.S. [E.S. Lyapin] Ljapin, "Semigroups" , Amer. Math. Soc. (1978) pp. Chapt. XII (Translated from Russian)
[a2] N.R. Reilly, "On the lattice of varieties of completely regular semigroups" S.M. Goberstein (ed.) P.M. Higgins (ed.) , Semigroups and Their Applications , Reidel (1987) pp. 153–167
How to Cite This Entry:
Variety of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variety_of_semi-groups&oldid=49130
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article