Semigroup with two elements

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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 nonisomorphic semigroups having two elements:

• O₂, the null semigroup of order two.
• LO₂, the left zero semigroup of order two.
• RO₂, the right zero semigroup of order two.
• ({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; this is also isomorphic to (Z₂, ·₂), the multiplicative group of {0,1} modulo 2.
• (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.