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| In [[mathematics]], a '''semigroup with two elements''' is a [[semigroup]] for which the [[cardinality]] of the [[underlying set]] is [[two]]. There are exactly five [[distinct]] [[isomorphic|nonisomorphic]] semigroups having two elements:
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| * ''O''<sub>2</sub>, the [[null semigroup]] of order two,
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| * ''LO''<sub>2</sub> and ''RO''<sub>2</sub>, the [[left zero semigroup]] of order two and [[right zero semigroup]] of order two, respectively,
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| * ({0,1}, ∧) (where "∧" is the [[logical connective]] "[[Logical conjunction|and]]"), the non-null semigroup with [[Null semigroup#Zero element in a semigroup|zero]] of order two, also a [[monoid]],
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| * (''Z''<sub>2</sub>, +<sub>2</sub>) (where ''Z''<sub>2</sub> = {0,1} and "+<sub>2</sub>" is "[[Modular addition|addition modulo 2]]"): the only [[group (mathematics)|group]] of order two.
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| The semigroups ''LO''<sub>2</sub> and ''RO''<sub>2</sub> are [[antiisomorphic]]. ''O''<sub>2</sub>, {{nowrap|({0,1}, ∧)}} and {{nowrap|(''Z''<sub>2</sub>, +<sub>2</sub>)}} are [[commutativity|commutative]], ''LO''<sub>2</sub> and ''RO''<sub>2</sub> are noncommutative. ''LO''<sub>2</sub>, ''RO''<sub>2</sub> and {{nowrap|({0,1}, ∧)}} are [[band (algebra)|band]]s and also inverse semigroups.
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| ==Determination of semigroups with two elements==
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| Choosing the set {{nowrap begin}}''A'' = { 1, 2 }{{nowrap end}} as the underlying set having two elements, sixteen [[binary operation]]s can be defined in ''A''. These operations are shown in the table below. In the table, a [[Matrix (mathematics)|matrix]] of the form
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| <center>
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| {| class="wikitable" border="1"
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| |-
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| | ''x''
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| | ''y''
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| |-
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| | ''z''
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| | ''t''
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| |}
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| </center>
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| indicates a binary operation on ''A'' having the following [[Cayley table]].
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| <center>
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| {| class="wikitable" border="1"
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| |-
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| !
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| ! 1
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| ! 2
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| |-
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| | '''1'''
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| | ''x''
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| | ''y''
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| |-
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| | '''2'''
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| | ''z''
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| | ''t''
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| |}
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| </center>
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| <center>
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| {| border="1" width="85%"
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| |+'''List of binary operations in { 1, 2 }'''
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| |-
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#87CEEB"
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| |-
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| | 1
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| | 1
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| |-
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| | 1
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#ADFF2F"
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| |-
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| | 1
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| | 1
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| |-
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| | 1
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| | 2
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| |}
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 1
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| | 1
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| |-
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| | 2
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#FFA500"
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| |-
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| | 1
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| | 1
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| |-
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| | 2
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| | 2
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| |}
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| |-
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| | '''Null semigroup O<sub>2</sub>'''
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| | '''≡ Semigroup ({0,1}, <math>\wedge</math>)'''
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| | 2·(1·2) = 2, (2·1)·2 = 1
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| | '''Left zero semigroup LO<sub>2</sub>'''
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| |-
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 1
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| | 2
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| |-
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| | 1
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#E0B0FF"
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| |-
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| | 1
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| | 2
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| |-
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| | 1
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| | 2
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#FF4500"
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| |-
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| | 1
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| | 2
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| |-
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| | 2
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#ADFF2F"
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| |-
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| | 1
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| | 2
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| |-
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| | 2
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| | 2
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| |}
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| |-
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| | 2·(1·2) = 1, (2·1)·2 = 2
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| | '''Right zero semigroup RO<sub>2</sub>'''
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| | '''≡ Group (Z<sub>2</sub>, +<sub>2</sub>)'''
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| | '''≡ Semigroup ({0,1}, <math>\wedge</math>)'''
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| |-
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 1
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| |-
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| | 1
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| | 1
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| |}
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| | align="center"|
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| {| class="wikitable" border="1" style="background:#FF4500"
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| |-
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| | 2
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| | 1
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| |-
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| | 1
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| | 2
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| |}
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| | align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 1
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| |-
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| | 2
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 1
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| |-
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| | 2
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| | 2
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| |}
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| |-
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| | 1·(1·2) = 2, (1·1)·2 = 1
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| | '''≡ Group (Z<sub>2</sub>, +<sub>2</sub>) '''
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| | 1·(1·1) = 1, (1·1)·1 = 2
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| | 1·(2·1) = 1, (1·2)·1 = 2
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| |-
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 2
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| |-
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| | 1
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 2
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| |-
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| | 1
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| | 2
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| |}
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| |align="center"|
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| {| class="wikitable" border="1"
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| |-
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| | 2
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| | 2
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| |-
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| | 2
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| | 1
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| |}
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| |align="center"|
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| {| class="wikitable" border="1" style="background:#87CEEB"
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| |-
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| | 2
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| | 2
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| |-
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| | 2
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| | 2
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| |}
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| |-
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| | 1·(1·1) = 2, (1·1)·1 = 1
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| | 1·(2·1) = 2, (1·2)·1 = 1
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| | 1·(1·2) = 2, (1·1)·2 = 1
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| | '''Null semigroup O<sub>2</sub>'''
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| |}
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| </center>
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| | |
| In this table:
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| *The semigroup ({0,1}, <math>\wedge</math>) denotes the two-element semigroup containing the [[Null semigroup#Zero element in a semigroup|zero element]] 0 and the [[unit element]] 1. The two binary operations defined by matrices in a green background are associative and pairing either with ''A'' creates a semigroup isomorphic to the semigroup ({0,1}, <math>\wedge</math>).
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| *The two binary operations defined by matrices in a blue background are associative and pairing either with ''A'' creates a semigroup isomorphic to the [[null semigroup]] ''O''<sub>2</sub> with two elements.
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| *The binary operation defined by the matrix in an orange background is associative and pairing it with ''A'' creates a semigroup. This is the [[Null semigroup#Left zero semigroup|left zero semigroup]] ''LO''<sub>2</sub>.
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| *The binary operation defined by the matrix in a purple background is associative and pairing it with ''A'' creates a semigroup. This is the [[Null semigroup#Right zero semigroup|right zero semigroup]] ''RO''<sub>2</sub>.
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| *The two binary operations defined by matrices in a red background are associative and pairing either with ''A'' creates a semigroup isomorphic to the [[group (mathematics)|group]] (Z<sub>2</sub>, +<sub>2</sub>).
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| *The remaining eight [[binary operation]]s defined by matrices in a white background are not [[associative]] and hence none of them create a semigroup when paired with ''A''.
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| ==The two-element semigroup ({0,1}, <math>\wedge</math>)==
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| The [[Cayley table]] for the semigroup ({0,1}, <math>\wedge</math>) is given below:
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| <center>
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| {| class="wikitable" border="1"
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| |-
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| ! <math>\wedge</math>
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| ! 0
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| ! 1
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| |-
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| | '''0'''
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| | 0
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| | 0
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| |-
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| | '''1'''
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| | 0
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| | 1
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| |}
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| </center>
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| This is the simplest non-trivial example of a semigroup that is not a group. This semigroup has an identity element, 1, making it a [[monoid]]. It is also commutative. It is not a group because the element 0 does not have an inverse, and is not even a cancellative semigroup because we cannot cancel the 0 in the equation 1·0 = 0·0.
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| This semigroup arises in various contexts. For instance, if we choose 1 to be the [[truth value]] "[[Truth|true]]" and 0 to be the [[truth value]] "[[False (logic)|false]]" and the operation to be the [[logical connective]] "[[Logical conjunction|and]]", we obtain this semigroup in [[logic]]. It is isomorphic to the monoid {0,1} under multiplication. It is also isomorphic to the semigroup
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| :<math>
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| S = \left\{
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| \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 1
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| \end{pmatrix},
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| \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 0
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| \end{pmatrix}
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| \right\}
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| </math>
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| under [[matrix multiplication]].<ref>{{PlanetMath reference|id=8498|title=Semigroup with two elements}}</ref>
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| ==The two-element semigroup (''Z''<sub>2</sub>,+<sub>2</sub>)==
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| The [[Cayley table]] for the semigroup (''Z''<sub>2</sub>,+<sub>2</sub>) is given below:
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| <center>
| |
| {| class="wikitable" border="1"
| |
| |-
| |
| !+<sub>2</sub>
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| ! 0
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| ! 1
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| |-
| |
| | '''0'''
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| | 0
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| | 1
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| |-
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| | '''1'''
| |
| | 1
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| | 0
| |
| |}
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| </center>
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| | |
| This group is isomorphic to the [[cyclic group]] Z<sub>2</sub> and the [[symmetric group]] S<sub>2</sub>.
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| | |
| ==Finite semigroups of higher orders==
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| Let ''A'' be the three-element set { 1, 2, 3 }. Altogether, a total of 3<sup>9</sup> = 19683 different binary operations can be defined on ''A''. It is impractical to prepare a list of all these binary operations, and determine those that make ''A'' nonisomorphic semigroups.
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| Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order. These have been applied to determine the nonisomorphic semigroups of order three or higher.<ref>{{Cite journal|last=Siniša Crvenkovič|coauthors=Ivan Stojmenovic|title=An algorithm for Cayley tables of algebras|publisher=Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Review of Research, Faculty of Science|volume=23|issue=2|pages=221–231}} [http://www.emis.de/journals/NSJOM/Papers/23_2/NSJOM_23_2_221_231.pdf] (Accessed on 9 May 2009)</ref><ref>{{Cite book|last=John A Hildebrant|title=Handbook of Finite Semigroup Programs|publisher=(Preprint)|year=2001}}[http://www.math.lsu.edu/~preprint/2001/jah2001a.pdf]</ref> In fact, 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism). The number of nonisomorphic semigroups with ''n'' elements, for ''n'' a nonnegative integer, is listed under {{OEIS2C|A027851}} in the [[On-Line Encyclopedia of Integer Sequences]]. {{OEIS2C|A001423}} lists the number of non-equivalent semigroups, and {{OEIS2C|A023814}} the number of associative binary operations, out of a total of ''n''<sup>''n''<sup>2</sup></sup>, determining a semigroup.
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| ==See also==
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| *[[Empty semigroup]]
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| *[[Trivial semigroup]] ([[Semigroup with one element]])
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| *[[Special classes of semigroups]]
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| | |
| ==References==
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| {{Reflist}}
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| {{Use dmy dates|date=September 2010}}
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| | |
| {{DEFAULTSORT:Semigroup With Two Elements}}
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| [[Category:Algebraic structures]]
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| [[Category:Semigroup theory]]
| |
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