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Nowhere commutative semigroup

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a nowhere commutative semigroup is a <a href="Semigroup" title="Semigroup">semigroup</a> S such that, for all a and b in S, if ab = ba then a = b.<a href="#cite_note-CP-1">[1]</a> A semigroup S is nowhere commutative <a href="If_and_only_if" title="If and only if">if and only if</a> any two elements of S are <a href="Inverse_semigroup" title="Inverse semigroup">inverses</a> of each other.<a href="#cite_note-CP-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Characterization_of_nowhere_commutative_semigroups">Characterization of nowhere commutative semigroups</h2></div>

Nowhere commutative semigroups can be <a href="Characterization_(mathematics)" title="Characterization (mathematics)">characterized</a> in several different ways. If S is a semigroup then the following statements are <a href="Logical_equivalence" title="Logical equivalence">equivalent</a>:<a href="#cite_note-Howie-2">[2]</a>

<ul><li>S is nowhere commutative.</li>

<li>S is a <a href="Rectangular_band" class="mw-redirect" title="Rectangular band">rectangular band</a> (in the sense in which the term is used by <a href="John_Mackintosh_Howie" title="John Mackintosh Howie">John Howie</a><a href="#cite_note-3">[3]</a>).</li>

Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in terms of the basic <a href="Binary_operation" title="Binary operation">binary operation</a> in the semigroup. The approach via the definition of nowhere commutative semigroups rectifies this defect.<a href="#cite_note-Howie-2">[2]</a>

To see that a nowhere commutative semigroup is a rectangular band, let S be a nowhere commutative semigroup. Using the defining properties of a nowhere commutative semigroup, one can see that for every a in S the <a href="Intersection_(mathematics)" class="mw-redirect" title="Intersection (mathematics)">intersection</a> of the <a href="Green's_relations" title="Green's relations">Green classes</a> Ra and La contains the unique element a. Let S / L be the family of L-classes in S and S / R be the family of R-classes in S. The mapping

defined by

is a <a href="Bijection" title="Bijection">bijection</a>. If the <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> (S / R) × (S / L) is made into a semigroup by furnishing it with the rectangular band multiplication, the map ψ becomes an <a href="Isomorphism" title="Isomorphism">isomorphism</a>. So S is isomorphic to a rectangular band.

Other claims of equivalences follow directly from the relevant definitions.

<a href="Special_classes_of_semigroups" title="Special classes of semigroups">Special classes of semigroups</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-CP-1">^ <a href="#cite_ref-CP_1-0">a</a> <a href="#cite_ref-CP_1-1">b</a> <a href="A._H._Clifford" class="mw-redirect" title="A. H. Clifford">A. H. Clifford</a>, <a href="G._B._Preston" class="mw-redirect" title="G. B. Preston">G. B. Preston</a> (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> (p.26). <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-0272-4</bdi>

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<li id="cite_note-Howie-2">^ <a href="#cite_ref-Howie_2-0">a</a> <a href="#cite_ref-Howie_2-1">b</a> <cite id="CITEREFJ._M._Howie1976" class="citation book cs1">J. M. Howie (1976). An Introduction to Semigroup Theory. LMS monographs. Vol. 7. Academic Press. p. 96.</cite>

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<cite id="CITEREFJ._M._Howie1976" class="citation book cs1">J. M. Howie (1976). An Introduction to Semigroup Theory. LMS monographs. Vol. 7. Academic Press. p. 3.</cite>

</li>

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