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Semigroup with two elements
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a semigroup with two elements is a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a> for which the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of the <a href="/facts/Underlying_set/TkuXca24">underlying set</a> is two. There are exactly five <a href="/facts/Isomorphic/pj8KgkxU">nonisomorphic</a> semigroups having two elements: </p> <ul><li>O2, the <a href="/facts/Null_semigroup/eyRwHwrM">null semigroup</a> of order two.</li> <li>LO2, the <a href="/facts/Left_zero_semigroup/eyRwHwrM">left zero semigroup</a> of order two.</li> <li>RO2, the <a href="/facts/Right_zero_semigroup/eyRwHwrM">right zero semigroup</a> of order two.</li> <li>({0,1}, ∧) (where "∧" is the <a href="/facts/Logical_connective/VkPhUs6M">logical connective</a> "<a href="/facts/Logical_conjunction/62uyyn1d">and</a>"), or equivalently the set {0,1} under multiplication: the only <a href="/facts/Semilattice/kW5po0rp">semilattice</a> with two elements and the only non-null semigroup with <a href="/facts/Null_semigroup/eyRwHwrM">zero</a> of order two, also a <a href="/facts/Monoid/6AHve7p8">monoid</a>, and ultimately the <a href="/facts/Two-element_Boolean_algebra/8nIWUpwd">two-element Boolean algebra</a>; this is also isomorphic to (Z2, ·2), the multiplicative group of {0,1} modulo 2.</li> <li>(Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "<a href="/facts/Exclusive_or/FHyr7hGQ">xor</a>"), or equivalently the set {−1,1} under multiplication: the only <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of order two.</li></ul> <p>The semigroups LO2 and RO2 are <a href="/facts/Antiisomorphic/buJ1P3PE">antiisomorphic</a>. O2, ({0,1}, ∧) and (Z2, +2) are <a href="/facts/Commutativity/WaYxJLxd">commutative</a>, and LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are <a href="/facts/Band_(algebra)/CdwD3R0Y">bands</a>. </p>