Q-guidance

2013-04-16T17:48:12Z

88.128.80.6: /* The Q Matrix */

{{Cleanup|reason=various, see talk|date=October 2012}}
In [[mathematics]], a [[semigroup]] is a [[nonempty set]] together with an [[associative]] [[binary operation]]. A '''special class of semigroups''' is a [[Class (set theory)|class]] of [[semigroup]]s satisfying additional [[property (philosophy)|properties]] or conditions. Thus the class of [[commutative]] semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup.
The class of [[Finite set|finite]] semigroups consists of those semigroups for which the [[underlying set]] has finite [[cardinality]]. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large variety of special classes of semigroups have been defined though not all of them have been studied equally intensively.

In the [[algebra]]ic [[theory]] of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of [[subset]]s of the [[underlying set]]. The underlying [[set (mathematics)|sets]] are not assumed to carry any other mathematical [[structure]]s like [[Partial order|order]] or [[topology]].

As in any algebraic theory, one of the main problems of the theory of semigroups is the [[classification theorems|classification]] of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the [[Group (mathematics)|group]].

A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.

== Notations ==

In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

<center>
{| class="wikitable" border="1"
|+Notations
|-
! Notation
! Meaning
|-
| ''S''
| Arbitrary semigroup
|-
| ''E''
| Set of idempotents in ''S''
|-
|''G''
|Group of units in ''S''
|-
|''X''
| Arbitrary set
|-
| ''a'', ''b'', ''c''
| Arbitrary elements of ''S''
|-
| ''x'', ''y'', ''z''
| Specific elements of ''S''
|-
| ''e'', ''f''. ''g''
| Arbitrary elements of ''E''
|-
| ''h''
| Specific element of ''E''
|-
|''l'', ''m, ''n''
| Arbitrary positive integers
|-
|''j'', ''k''
| Specific positive integers
|-
| 0
| Zero element of ''S''
|-
| 1
| Identity element of ''S''
|-
|''S''1
| ''S'' if 1 ∈ ''S''; ''S'' ∪ { 1 } if 1 ∉ ''S''
|-
| ''L'', ''R'', ''H'', ''D'', ''J''
| Green's relations
|-
| ''L''''a'', ''R''''a'', ''H''''a'', ''D''''a'', ''J''''a''
| Green classes containing ''a''
|-
| ''a'' ≤''L'' ''b'' ''a'' ≤''R'' ''b'' ''a'' ≤''H'' ''b''
| ''S''1''a'' ⊆ ''S''1''b'' ''aS''1 ⊆ ''bS''1 ''S''1''a'' ⊆ ''S''1''b'' and ''aS''1 ⊆ ''bS''1
|-
|}
</center>

== List of special classes of semigroups ==

<center>

{| class="wikitable sortable" border="1" width="80%"
|+List of special classes of semigroups
|-
! Terminology
! class="unsortable" |Defining property
! Reference(s)
|-
|[[Finite set|Finite]] semigroup
|
*''S'' is a [[finite set]].
|
|-
|[[Empty semigroup]]
|
*''S'' = <math>\emptyset</math>
|
|-
|[[Trivial semigroup]]
|
*Cardinality of ''S'' is 1.
|
|-
|[[Monoid]]
|
*1 ∈ ''S''
|[[#Gril|Gril]] p. 3
|-
| [[Band (mathematics)|Band]] (Idempotent semigroup)
|
*''a''2 = ''a''
|[[#C&P|C&P]] p. 4
|-
| [[Semilattice]]
|
*''a''2 = ''a''
*''ab '' = ''ba ''
|[[#C&P|C&P]] p. 24
|-
| [[Commutative]] semigroup
|
*''ab'' = '' ba ''
| [[#C&P|C&P]] p. 3
|-
|[[Archimedean property|Archimedean]] commutative semigroup
|
*''ab'' = ''ba''
*There exists ''x'' and ''k'' such that ''a'' = ''xb''''k''.
|[[#C&P|C&P]] p. 131
|-
|[[Nowhere commutative semigroup]]
|
*''ab'' = ''ba''   ⇒   ''a'' = ''b''
|[[#C&P|C&P]] p. 26
|-
|Left weakly commutative
|
*There exist ''x'' and ''k'' such that (''ab'')''k'' = ''bx''.
|[[#Nagy|Nagy]] p. 59
|-
|Right weakly commutative
|
*There exist ''x'' and ''k'' such that (''ab'')''k'' = ''xb''.
|[[#Nagy|Nagy]] p. 59
|-
|Weakly commutative
|
* There exist ''x'' and ''j'' such that (''ab'')''j'' = ''bx''.
* There exist ''y'' and ''k'' such that (''ab'')''k'' = ''yb''.
|[[#Nagy|Nagy]] p. 59
|-
|Conditionally commutative semigroup
|
*If ''ab'' = ''ba'' then ''axb'' = ''bxa'' for all ''x''.
|[[#Nagy|Nagy]] p. 77
|-
|''R''-commutative semigroup
|
*''ab'' ''R'' ''ba''
|[[#Nagy|Nagy]] p. 69–71
|-
|''RC''-commutative semigroup
|
*''R''-commutative and conditionally commutative
|[[#Nagy|Nagy]] p. 93–107
|-
|''L''-commutative semigroup
|
*''ab'' ''L'' ''ba''
|[[#Nagy|Nagy]] p. 69–71
|-
|''LC''-commutative semigroup
|
*''L''-commutative and conditionally commutative
|[[#Nagy|Nagy]] p. 93–107

|-
|''H''-commutative semigroup
|
*''ab'' ''H'' ''ba''
|[[#Nagy|Nagy]] p. 69–71
|-
|Quasi-commutative semigroup
|
*''ab'' = (''ba'')''k'' for some ''k''.
|[[#Nagy|Nagy]] p. 109
|-
|Right commutative semigroup
|
*''xab'' = ''xba''
|[[#Nagy|Nagy]] p. 137
|-
|Left commutative semigroup
|
*''abx'' = ''bax''
|[[#Nagy|Nagy]] p. 137
|-
|Externally commutative semigroup
|
*''axb'' = ''bxa''
|[[#Nagy|Nagy]] p. 175
|-
|Medial semigroup
|
*''xaby'' = ''xbay''
|[[#Nagy|Nagy]] p. 119
|-
|E-''k'' semigroup (''k'' fixed)
|
*(''ab'')''k'' = ''a''''k''''b''''k''
|[[#Nagy|Nagy]] p. 183
|-
|[[Exponential]] semigroup
|
*(''ab'')''m'' = ''a''''m''''b''''m'' for all ''m''
|[[#Nagy|Nagy]] p. 183
|-
|WE-''k'' semigroup (''k'' fixed)
|
*There is a positive integer ''j'' depending on the couple (a,b) such that (''ab'')''k''+''j'' = ''a''''k''''b''''k'' (''ab'')''j'' = (''ab'')''j''''a''''k''''b''''k''
|[[#Nagy|Nagy]] p. 199
|-
|Weakly [[exponential]] semigroup
|
*WE-''m'' for all ''m''
|[[#Nagy|Nagy]] p. 215
|-
|[[Cancellative semigroup]]
|
*''ax = ay ''   ⇒   ''x = y''
*''xa = ya ''   ⇒   ''x = y''
|[[#C&P|C&P]] p. 3
|-
|[[Cancellative semigroup|Right cancellative semigroup]]
|
*''xa = ya ''   ⇒   ''x = y''
|[[#C&P|C&P]] p. 3
|-
|[[Cancellative semigroup|Left cancellative semigroup]]
|
*''ax = ay ''   ⇒   ''x = y''
|[[#C&P|C&P]] p. 3
|-
|''E''-[[inversive]] semigroup
|
*There exists ''x'' such that ''ax'' ∈ ''E''.
|[[#C&P|C&P]] p. 98
|-
|[[Regular semigroup]]
|
*There exists ''x'' such that ''axa'' =''a''.
|[[#C&P|C&P]] p. 26
|-
|Intra-regular semigroup
|
*There exist ''x'' and ''y'' such that ''xa''2''y'' = ''a''.
|[[#C&P|C&P]] p. 121
|-
|Left regular semigroup
|
*There exists ''x'' such that ''xa''2 = ''a''.
|[[#C&P|C&P]] p. 121
|-
|Right regular semigroup
|
*There exists ''x'' such that ''a''2''x'' = ''a''.
|[[#C&P|C&P]] p. 121
|-
|[[Completely regular semigroup]]
|
*''H''''a'' is a group.
|[[#Gril|Gril]] p. 75
|-
|(inverse) [[Clifford semigroup]]
|
*A regular semigroup in which all idempotents are central.
|[[#Pet|Petrich]] p. 65
|-
|''k''-regular semigroup (''k'' fixed)
|
*There exists ''x'' such that ''a''''k''''xa''''k'' = ''a''''k''.
|[[#Hari|Hari]]
|-
|Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup)
|
*There exists ''k'' and ''x'' (depending on ''a'') such that ''a''''k''''xa''''k'' = ''a''''k''.
|[[#Edwa|Edwa]] [[#Shum|Shum]] [[#Higg|Higg]] p. 49
|-
|Quasi-periodic semigroup, [[epigroup]], group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see [[#Kela|Kela]] for a list)
|
*There exists ''k'' (depending on ''a'') such that ''a''''k'' belongs to a [[subgroup]] of ''S''
|[[#Kela|Kela]] [[#Gril|Gril]] p. 110 [[#Higg|Higg]] p. 4
|-
|Primitive semigroup
|
*If ''0'' ≠ ''e'' and ''f'' = ''ef'' = ''fe'' then ''e'' = ''f''.
|[[#C&P|C&P]] p. 26
|-
|Unit regular semigroup
|
*There exists ''u'' in ''G'' such that ''aua'' = ''a''.
|[[#Tvm|Tvm]]
|-
|Strongly unit regular semigroup
|
*There exists ''u'' in ''G'' such that ''aua'' = ''a''.
*''e D f'' ⇒ ''f'' = ''v''−1''ev'' for some ''v'' in ''G''.
|[[#Tvm|Tvm]]
|-
|[[Orthodox semigroup]]
|
*There exists ''x'' such that ''axa'' = ''a''.
*''E'' is a subsemigroup of ''S''.
|[[#Gril|Gril]] p. 57 [[#Howi|Howi]] p. 226
|-
|[[Inverse semigroup]]
|
*There exists unique ''x'' such that ''axa'' = ''a'' and ''xax'' = ''x''.
|[[#C&P|C&P]] p. 28
|-
|Left inverse semigroup (''R''-unipotent)
|
*''R''''a'' contains a unique ''h''.
|[[#Gril|Gril]] p. 382
|-
|Right inverse semigroup (''L''-unipotent)
|
*''L''''a'' contains a unique ''h''.
|[[#Gril|Gril]] p. 382
|-
|Locally inverse semigroup (Pseudoinverse semigroup)
|
*There exists ''x'' such that ''axa'' = ''a''.
*''E'' is a pseudosemilattice.
|[[#Gril|Gril]] p. 352
|-
|''M''-inversive semigroup
|
*There exist ''x'' and ''y'' such that ''baxc'' = ''bc'' and ''byac'' = ''bc''.
|[[#C&P|C&P]] p. 98
|-
|Pseudoinverse semigroup (Locally inverse semigroup)
|
*There exists ''x'' such that ''axa'' = ''a''.
* ''E'' is a pseudosemilattice.
|[[#Gril|Gril]] p. 352
|-
|Abundant semigroups
|
*The classes ''L''*''a'' and ''R''*''a'', where ''a'' ''L''* ''b'' if ''ac'' = ''ad'' ⇔ ''bc'' = ''bd'' and ''a'' ''R''* ''b'' if ''ca'' = ''da'' ⇔ ''cb'' = ''db'', contain idempotents.
|[[#Chen|Chen]]
|-
|Rpp-semigroup (Right principal projective semigroup)
|
*The class ''L''*''a'', where ''a'' ''L''* ''b'' if ''ac'' = ''ad'' ⇔ ''bc'' = ''bd'', contains at least one idempotent.
|[[#Shum|Shum]]
|-
|Lpp-semigroup (Left principal projective semigroup)
|
*The class ''R''*''a'', where ''a'' ''R''* ''b'' if ''ca'' = ''da'' ⇔ ''cb'' = ''db'', contains at least one idempotent.
|[[#Shum|Shum]]
|-
|[[Null semigroup]] ([[Zero semigroup]])
|
*0 ∈ ''S''
*''ab'' = 0
|[[#C&P|C&P]] p. 4
|-
|[[Zero semigroup]] ([[Null semigroup]])
|
*0 ∈ ''S''
*''ab'' = 0
|[[#C&P|C&P]] p. 4
|-
|[[Left zero semigroup]]
|
*''ab'' = ''a''
|[[#C&P|C&P]] p. 4
|-
|[[Right zero semigroup]]
|
*''ab'' = ''b''
|[[#C&P|C&P]] p. 4
|-
|Unipotent semigroup
|
*''E'' is singleton.
|[[#C&P|C&P]] p. 21
|-
|Left reductive semigroup
|
*If ''xa'' = ''xb'' for all ''x'' implies ''a'' = ''b''.
|[[#C&P|C&P]] p. 9
|-
|Right reductive semigroup
|
*If ''ax'' = ''bx'' for all ''x'' implies ''a'' = ''b''.
|[[#C&P|C&P]] p. 4
|-
|Reductive semigroup
|
*If ''xa'' = ''xb'' for all ''x'' implies ''a'' = ''b''.
*If ''ax'' = ''bx'' for all ''x'' implies ''a'' = ''b''.
|[[#C&P|C&P]] p. 4
|-
|Separative semigroup
|
*''ab'' = ''a''2 = ''b''2   ⇒   ''a'' = ''b''
|[[#C&P|C&P]] p. 130–131
|-
|Reversible semigroup
|
*''Sa'' ∩ ''Sb'' ≠ Ø
*''aS'' ∩ ''bS'' ≠ Ø
|[[#C&P|C&P]] p. 34
|-
|Right reversible semigroup
|
*''Sa'' ∩ ''Sb'' ≠ Ø
|[[#C&P|C&P]] p. 34
|-
|Left reversible semigroup
|
*''aS'' ∩ ''bS'' ≠ Ø
|[[#C&P|C&P]] p. 34
|-
|[[Aperiodic semigroup]] 
|
* There exists ''k'' (depending on ''a'') such that ak = ak+1
|[[#KKM|KKM]] p. 29 
|-
|ω-semigroup
|
*E is countable descending chain under the order ''a'' ≤''H'' ''b''
|[[#Gril|Gril]] p. 233–238
|-
|Left Clifford semigroup (LC-semigroup)
|
*''aS'' ⊆ ''Sa''
|[[#Shum|Shum]]
|-
|Right Clifford semigroup (RC-semigroup)
|
*''Sa'' ⊆ ''aS''
|[[#Shum|Shum]]
|-
|LC-semigroup (Left Clifford semigroup)
|
*''aS'' ⊆ ''Sa''
|[[#Shum|Shum]]
|-
|RC-semigroup (Right Clifford semigroup)
|
*''Sa'' ⊆ ''aS''
|[[#Shum|Shum]]
|-
|Orthogroup
|
*''H''''a'' is a group.
*''E'' is a subsemigroup of ''S''
|[[#Shum|Shum]]
|-
|Complete commutative semigroup
|
*''ab'' = ''ba''
* ''a''''k'' is in a subgroup of ''S'' for some ''k''.
*Every nonempty subset of ''E'' has an infimum.
|[[#Gril|Gril]] p. 110
|-
|Nilsemigroup
|
*0 ∈ ''S''
*''a''''k'' = 0 for some ''k''.
|[[#Gril|Gril]] p. 99
|-
|Elementary semigroup
|
*''ab'' = ''ba''
*''S'' = ''G'' ∪ ''N'' where ''G'' is a group, ''N'' is a nilsemigroup or a one-element semigroup.
*''N'' is ideal of ''S''.
*Identity of ''G'' is 1 of ''S'' and zero of ''N'' is 0 of ''S''.
|[[#Gril|Gril]] p. 111
|-
|''E''-unitary semigroup
|
*There exists unique ''x'' such that ''axa'' = ''a'' and ''xax'' = ''x''.
*''ea'' = ''e''   ⇒   ''a'' ∈ ''E''
|[[#Gril|Gril]] p. 245
|-
|Finitely presented semigroup
|
*''S'' has a presentation ( ''X''; ''R'' ) in which ''X'' and ''R'' are finite.
|[[#Gril|Gril]] p. 134
|-
|Fundamental semigroup
|
*Equality on ''S'' is the only congruence contained in ''H''.
|[[#Gril|Gril]] p. 88
|-
|Idempotent generated semigroup
|
*''S'' is equal to the semigroup generated by ''E''.
|[[#Gril|Gril]] p. 328
|-
|Locally finite semigroup
|
*Every finitely generated subsemigroup of ''S'' is finite.
|[[#Gril|Gril]] p. 161
|-
|''N''-semigroup
|
*''ab'' = ''ba''
*There exists ''x'' and a positive integer ''n'' such that ''a'' = ''xb''n.
*''ax = ay ''   ⇒   ''x = y''
*''xa = ya ''   ⇒   ''x = y''
*''E'' = Ø
|[[#Gril|Gril]] p. 100
|-
|''L''-unipotent semigroup (Right inverse semigroup)
|
*''L''''a'' contains a unique ''e''.
|[[#Gril|Gril]] p. 362
|-
|''R''-unipotent semigroup (Left inverse semigroup)
|
*''R''''a'' contains a unique ''e''.
|[[#Gril|Gril]] p. 362
|-
|Left simple semigroup
|
*''L''''a'' = ''S''
|[[#Gril|Gril]] p. 57
|-
|Right simple semigroup
|
*''R''''a'' = ''S''
|[[#Gril|Gril]] p. 57
|-
|Subelementary semigroup
|
*''ab'' = ''ba''
* ''S'' = ''C'' ∪ ''N'' where ''C'' is a cancellative semigroup, ''N'' is a nilsemigroup or a one-element semigroup.
*''N'' is ideal of ''S''.
*Zero of ''N'' is 0 of ''S''.
*For ''x'', ''y'' in ''S'' and ''c'' in ''C'', ''cx'' = ''cy'' implies that ''x'' = ''y''.
|[[#Gril|Gril]] p. 134
|-
|Symmetric semigroup ([[Transformation semigroup|Full transformation semigroup]])
|
*Set of all mappings of ''X'' into itself with composition of mappings as binary operation.
|[[#C&P|C&P]] p. 2
|-
|Weakly reductive semigroup
|
*If ''xz'' = ''yz'' and ''zx'' = ''zy'' for all ''z'' in ''S'' then ''x'' = ''y''.
|[[#C&P|C&P]] p. 11
|-
|Right unambiguous semigroup
|
*If ''x'', ''y'' ≥''R'' ''z'' then ''x'' ≥''R'' ''y'' or ''y'' ≥''R'' ''x''.
|[[#Gril|Gril]] p. 170
|-
|Left unambiguous semigroup
|
*If ''x'', ''y'' ≥''L'' ''z'' then ''x'' ≥''L'' ''y'' or ''y'' ≥''L'' ''x''.
|[[#Gril|Gril]] p. 170
|-
|Unambiguous semigroup
|
*If ''x'', ''y'' ≥''R'' ''z'' then ''x'' ≥''R'' ''y'' or ''y'' ≥''R'' ''x''.
*If ''x'', ''y'' ≥''L'' ''z'' then ''x'' ≥''L'' ''y'' or ''y'' ≥''L'' ''x''.
|[[#Gril|Gril]] p. 170
|-
|Left 0-unambiguous
|
*0∈ ''S''
*0 ≠ ''x'' ≤''L'' ''y'', ''z''   ⇒   ''y'' ≤''L'' ''z'' or ''z'' ≤''L'' ''y''
|[[#Gril|Gril]] p. 178
|-
|Right 0-unambiguous
|
*0∈ ''S''
*0 ≠ ''x'' ≤''R'' ''y'', ''z''   ⇒   ''y'' ≤''L'' ''z'' or ''z'' ≤''R'' ''y''
|[[#Gril|Gril]] p. 178
|-
|0-unambiguous semigroup
|
*0∈ ''S''
*0 ≠ ''x'' ≤''L'' ''y'', ''z''   ⇒   ''y'' ≤''L'' ''z'' or ''z'' ≤''L'' ''y''
*0 ≠ ''x'' ≤''R'' ''y'', ''z''   ⇒   ''y'' ≤''L'' ''z'' or ''z'' ≤''R'' ''y''
|[[#Gril|Gril]] p. 178
|-
|Left Putcha semigroup
|
*''a'' ∈ ''bS''1   ⇒   ''a''''n'' ∈ ''b''2''S''1 for some ''n''.
|[[#Nagy|Nagy]] p. 35
|-
|Right Putcha semigroup
|
*''a'' ∈ ''S''1''b''   ⇒   ''a''''n'' ∈ ''S''1''b''2 for some ''n''.
|[[#Nagy|Nagy]] p. 35
|-
|Putcha semigroup
|
*''a'' ∈ ''S''1''b'' ''S''1   ⇒   ''a''''n'' ∈ ''S''1''b''2''S''1 for some positive integer ''n''
|[[#Nagy|Nagy]] p. 35
|-
|Bisimple semigroup (''D''-simple semigroup)
|
*''D''''a'' = ''S''
|[[#C&P|C&P]] p. 49
|-
|0-bisimple semigroup
|
*0 ∈ ''S''
* ''S'' - {0} is a ''D''-class of ''S''.
|[[#C&P|C&P]] p. 76
|-
|Completely simple semigroup
|
*There exists no ''A'' ⊆ ''S'', ''A'' ≠ ''S'' such that ''SA'' ⊆ ''A'' and ''AS'' ⊆ ''A''.
*There exists ''h'' in ''E'' such that whenever ''hf'' = ''f'' and ''fh'' = '' f '' we have ''h'' = ''f''.
|[[#C&P|C&P]] p. 76
|-
|Completely 0-simple semigroup
|
*0 ∈ ''S''
* ''S''2 ≠ 0
* If ''A'' ⊆ ''S'' is such that ''AS'' ⊆ ''A'' and ''SA'' ⊆ ''A'' then ''A'' = 0.
* There exists non-zero ''h'' in ''E'' such that whenever ''hf'' = ''f'', ''fh'' = ''f'' and ''f'' ≠ 0 we have ''h'' = ''f''.
|[[#C&P|C&P]] p. 76
|-
|''D''-simple semigroup (Bisimple semigroup)
|
*''D''''a'' = ''S''
|[[#C&P|C&P]] p. 49
|-
|Semisimple semigroup
|
*Let ''J''(''a'') = ''S''1''aS''1, ''I''(''a'') = ''J''(''a'') − ''J''''a''. Each Rees factor semigroup ''J''(''a'')/''I''(''a'') is 0-simple or simple.
|[[#C&P|C&P]] p. 71–75
|-
|Simple semigroup
|
* ''Ja'' = ''S''. (There exists no ''A'' ⊆ ''S'', ''A'' ≠ ''S'' such that ''SA'' ⊆ ''A'' and ''AS'' ⊆ ''A''.)
|[[#C&P|C&P]] p. 5 [[#Higg|Higg]] p. 16
|-
|0-simple semigroup
|
*0 ∈ ''S''
* ''S''2 ≠ 0
* If ''A'' ⊆ ''S'' is such that ''AS'' ⊆ ''A'' and ''SA'' ⊆ ''A'' then ''A'' = 0.
|[[#C&P|C&P]] p. 67
|-
|Left 0-simple semigroup
|
*0 ∈ ''S''
*''S''2 ≠ 0
* If ''A'' ⊆ ''S'' is such that ''SA'' ⊆ ''A'' then ''A'' = 0.
|[[#C&P|C&P]] p. 67
|-
|Right 0-simple semigroup
|
*0 ∈ ''S''
* ''S''2 ≠ 0
* If ''A'' ⊆ ''S'' is such that ''AS'' ⊆ ''A'' then ''A'' = 0.
|[[#C&P|C&P]] p. 67
|-
|[[Cyclic semigroup]] ([[Monogenic semigroup]])
|
*''S'' = { ''w'', ''w''2, ''w''3, ... } for some ''w'' in ''S''
|[[#C&P|C&P]] p. 19
|-
| [[Monogenic semigroup]] ([[Cyclic semigroup]])
|
*''S'' = { ''w'', ''w''2, ''w''3, ... } for some ''w'' in ''S''
|[[#C&P|C&P]] p. 19
|-
|Periodic semigroup
|
*{ ''a'', ''a''2, ''a''3, ... } is a finite set.
|[[#C&P|C&P]] p. 20
|-
|[[Bicyclic semigroup]]
|
*1 ∈ S
* ''S'' generated by { ''x''1, ''x''2 } with ''x''1''x''2 = 1.
|[[#C&P|C&P]] p. 43–46
|-
|[[Full transformation semigroup]] ''T''''X'' (Symmetric semigroup)
|
*[[Set (mathematics)|Set]] of all [[Map (mathematics)|mappings]] of ''X'' into itself with [[Composition of functions|composition of mappings]] as [[binary operation]].
|[[#C&P|C&P]] p. 2
|-
|Rectangular semigroup
|
*Whenever three of ''ax'', ''ay'', ''bx'', ''by'' are equal, all four are equal.
|[[#C&P|C&P]] p. 97
|-
|[[Symmetric inverse semigroup]] ''I''''X''
|
*The semigroup of [[bijection|one-to-one]] [[partial function|partial transformations]] of ''X''.
|[[#C&P|C&P]] p. 29
|-
|[[Brandt semigroup]]
|
*0 ∈ ''S''
* ( ''ac'' = ''bc'' ≠ 0 or ''ca'' = ''cb'' ≠ 0 )   ⇒   ''a'' = ''b''
* ( ''ab'' ≠ 0 and ''bc'' ≠ 0 )   ⇒   ''abc'' ≠ 0
* If ''a'' ≠ 0 there exist unique ''x'', ''y'', ''z'', such that ''xa'' = ''a'', ''ay'' = ''a'', ''za'' = ''y''.
* ( ''e'' ≠ 0 and ''f'' ≠ 0 )   ⇒   ''eSf '' ≠ 0.
|[[#C&P|C&P]] p. 101
|-
|[[Free semigroup]] ''F''''X''
|
*Set of finite sequences of elements of ''X'' with the operation ( ''x''1, ..., ''x''m ) ( ''y''1, ..., ''y''n ) = ( ''x''1, ..., ''x''m, ''y''1, ..., ''y''n )
|[[#Gril|Gril]] p. 18
|-
|Rees [[Matrix (mathematics)|matrix]] semigroup
|
*''G''0 a group ''G'' with 0 adjoined.
*''P'' : Λ × ''I'' → ''G''0 a map.
* Define an operation in ''I'' × ''G''0 × Λ by ( ''i'', ''g'', λ ) ( ''j'', ''h'', μ ) = ( ''i'', ''g'' P( λ, ''j'' ) ''h'', μ ).
* ( ''I'', ''G''0, Λ )/( ''I'' × { 0 } × Λ ) is the Ress matrix semigroup ''M''0 ( ''G''0; ''I, Λ ; ''P'' ).
|[[#C&P|C&P]] p.88
|-
|Semigroup of [[linear transformation]]s
|
*Semigroup of [[linear transformation]]s of a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''F'' under [[composition of functions]].
|[[#C&P|C&P]] p.57
|-
|Semigroup of [[binary relation]]s ''B''''X''
|
*Set of all [[binary relation]]s on ''X'' under [[composition of relations|composition]]
|[[#C&P|C&P]] p.13
|-
|[[Numerical semigroup]]
|
*0 ∈ ''S'' ⊆ ''N'' = { 0,1,2, ... } under + .
*''N'' - ''S'' is finite
|[[#Delg|Delg]]
|-
|[[Semigroup with involution]] (*-semigroup)
|
*There exists a unary operation ''a'' → ''a''* in ''S'' such that ''a''** = ''a'' and (''ab'')* = ''b''*''a''*.
|[[#Howi|Howi]]
|-
|*-semigroup ([[Semigroup with involution]])
|
*There exists a unary operation ''a'' → ''a''* in ''S'' such that ''a''** = ''a'' and (''ab'')* = ''b''*''a''*.
|[[#Howi|Howi]]
|-
|Baer–Levi semigroup
|
*Semigroup of one-to-one transformations ''f'' of ''X'' such that ''X'' − ''f'' ( ''X'' ) is infinite.
|[[#C&P II|C&P II]] Ch.8
|-
|''U''-semigroup
|
*There exists a unary operation ''a'' → ''a''’ in ''S'' such that ( ''a''’)’ = ''a''.
|[[#Howi|Howi]] p.102
|-
|''I''-semigroup
|
*There exists a unary operation ''a'' → ''a''’ in ''S'' such that ( ''a''’)’ = ''a'' and ''aa''’''a'' = ''a''.
|[[#Howi|Howi]] p.102
|-
|[[Semiband]]
|
*A regular semigroup generated by its idempotents.
|[[#Howi|Howi]] p.230
|-
|[[Group (mathematics)|Group]]
|
*There exists ''h'' such that for all a, ''ah'' = ''ha'' = ''a''.
*There exists ''x'' (depending on ''a'') such that ''ax'' = ''xa'' = ''h''.
|
|}

</center>

==References==

{|
|-valign="top"
| [C&P]
| {{Anchor|C&P}}A H Clifford, G B Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). [[American Mathematical Society]]. ISBN 978-0-8218-0272-4
|-valign="top"
|[C&P II]  
|{{Anchor|C&P II}}A H Clifford, G B Preston (1967). ''The Algebraic Theory of Semigroups Vol. II'' (Second Edition). [[American Mathematical Society]]. ISBN 0-8218-0272-0
|-valign="top"
| [Chen] 
| {{Anchor|Chen}}Hui Chen (2006), "Construction of a kind of abundant semigroups", ''Mathematical Communications'' ('''11'''), 165–171 (Accessed on 25 April 2009)
|-valign="top"
|[Delg]
|{{Anchor|Delg}}M Delgado, ''et al.'', ''Numerical semigroups'', [http://www.gap-system.org/Manuals/pkg/numericalsgps/doc/manual.pdf] (Accessed on 27 April 2009)
|-valign="top"
|[Edwa]
|{{Anchor|Edwa}}P Edwards (1983), "Eventually regular semigroups", ''Bulletin of Australian Mathematical Society'' '''28''', 23–38
|-valign="top"
|[Gril]
|{{Anchor|Gril}}P A Grillet (1995). ''Semigroups''. [[CRC Press]]. ISBN 978-0-8247-9662-4
|-valign="top"
|[Hari]
|{{Anchor|Hari}}K S Harinath (1979), "Some results on ''k''-regular semigroups", ''Indian Journal of Pure and Applied Mathematics'' '''10'''(11), 1422–1431
|-valign="top"
|[Howi]
|{{Anchor|Howi}}J M Howie (1995), ''Fundamentals of Semigroup Theory'', [[Oxford University Press]]
|-valign="top"
|[Nagy]
|{{Anchor|Nagy}}Attila Nagy (2001). ''Special Classes of Semigroups''. [[Springer Science+Business Media|Springer]]. ISBN 978-0-7923-6890-8
|-valign="top"
|[Pet]
|{{Anchor|Pet}} M Petrich, N R Reilly (1999). ''Completely regular semigroups''. [[John Wiley & Sons]]. ISBN 978-0-471-19571-9
|-valign="top"
|[Shum]    
|{{Anchor|Shum}}K P Shum "Rpp semigroups, its generalizations and special subclasses" in ''Advances in Algebra and Combinatorics'' edited by K P Shum et al. (2008), [[World Scientific]], ISBN 981-279-000-4 (pp. 303–334)
|-valign="top"
|[Tvm]
|{{Anchor|Tvm}}''Proceedings of the International Symposium on Theory of Regular Semigroups and Applications'', [[University of Kerala]], [[Thiruvananthapuram]], [[India]], 1986
|-valign="top"
|[Kela]
|{{Anchor|Kela}}A. V. Kelarev, ''Applications of epigroups to graded ring theory'', Semigroup Forum, Volume 50, Number 1 (1995), 327-350 {{doi|10.1007/BF02573530}} 
|-valign="top"
|[KKM]
|{{Anchor|KKM}}Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), ''Monoids, Acts and Categories: with Applications to Wreath Products and Graphs'', Expositions in Mathematics '''29''', Walter de Gruyter, Berlin, ISBN 978-3-11-015248-7.
|-valign="top"
|[Higg]
|{{Anchor|Higg}} {{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5}}
|}

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[[Category:Semigroup theory]]

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