University of Arkansas, Fayetteville ScholarWorks@UARK Theses and Dissertations 12-2012 On the Representation of Inverse Semigroups by Difunctional Relations Nathan Bloomfield University of Arkansas, Fayetteville Follow this and additional works at: http://scholarworks.uark.edu/etd Part of the Mathematics Commons Recommended Citation Bloomfield, Nathan, "On the Representation of Inverse Semigroups by Difunctional Relations" (2012). Theses and Dissertations. 629. http://scholarworks.uark.edu/etd/629 This Dissertation is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ScholarWorks@UARK. For more information, please contact
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[email protected]. ON THE REPRESENTATION OF INVERSE SEMIGROUPS BY DIFUNCTIONAL RELATIONS On the Representation of Inverse Semigroups by Difunctional Relations A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by Nathan E. Bloomfield Drury University Bachelor of Arts in Mathematics, 2007 University of Arkansas Master of Science in Mathematics, 2011 December 2012 University of Arkansas Abstract A semigroup S is called inverse if for each s 2 S, there exists a unique t 2 S such that sts = s and tst = t. A relation σ ⊆ X × Y is called full if for all x 2 X and y 2 Y there exist x0 2 X and y0 2 Y such that (x; y0) and (x0; y) are in σ, and is called difunctional if σ satisfies the equation σσ-1σ = σ. Inverse semigroups were introduced by Wagner and Preston in 1952 [55] and 1954 [38], respectively, and difunctional relations were introduced by Riguet in 1948 [39].