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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 nonisomorphic semigroups having two elements: * O2, the null semigroup of order two, * LO2 and RO2, the left zero semigroup of order two and right zero semigroup of order two, respectively, * ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, * (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: t

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rdfs:label	Semigroup with two elements (en)
rdfs:comment	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 nonisomorphic semigroups having two elements: * O2, the null semigroup of order two, * LO2 and RO2, the left zero semigroup of order two and right zero semigroup of order two, respectively, * ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, * (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: t (en)
dcterms:subject	Semigroup theory Algebraic structures
Wikipage page ID	22717084 (xsd:integer)
Wikipage revision ID	1121809382 (xsd:integer)
Link from a Wikipage to another Wikipage	Underlying set Inverse semigroup Null semigroup Semigroup theory Commutativity Mathematics Matrix (mathematics) Matrix multiplication Monoid Antiisomorphic Linear order Logic Logical conjunction Complemented lattice Empty semigroup Band (mathematics) Algebraic structures Truth Truth value Distributive lattice Lattice (order) Semigroup Semigroup with three elements Cyclic group Exclusive or False (logic) Band (algebra) Cardinality Cayley table Logical connective Group (mathematics) Isomorphic Special classes of semigroups Binary operation Distinct (mathematics) Idempotent Associative On-Line Encyclopedia of Integer Sequences Semilattice Symmetric group Two-element Boolean algebra Trivial semigroup Right zero semigroup Left zero semigroup Unit element Semigroup with one element
sameAs	Semigroup with two elements Semigroup with two elements Semigroup with two elements Semigroup with two elements
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has abstract	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 nonisomorphic semigroups having two elements: * O2, the null semigroup of order two, * LO2 and RO2, the left zero semigroup of order two and right zero semigroup of order two, respectively, * ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, * (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: the only group of order two. The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands and also inverse semigroups. (en)
gold:hypernym	Semigroup
prov:wasDerivedFrom	wikipedia-en:Semigroup_with_two_elements?oldid=1121809382&ns=0
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foaf:isPrimaryTopicOf	wikipedia-en:Semigroup_with_two_elements
is Link from a Wikipage to another Wikipage of	Aperiodic semigroup Empty semigroup Semigroup Semigroup with three elements Trivial semigroup Semigroup with 2 elements Semigroups with two elements
is Wikipage redirect of	Semigroup with 2 elements Semigroups with two elements
is foaf:primaryTopic of	wikipedia-en:Semigroup_with_two_elements

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