a notion of division for concrete monoids
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a notion of division for concrete monoids
Electronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidscrete MonoidsAndrew SolomonSchool of Mathematics and StatisticsThe University of SydneyNSW 2006, AustraliaAbstractA concrete monoid over a category c is a subset of the endomorphisms of an object of c. containing the identity and closed under composition. To contrast, an abstract monoid is just a on a notion of division for concrete monoidse object category.There is a natural notion of division between concrete monoids distinct from the usual division of abstract monoids. This concrete da notion of division for concrete monoids
ivision is identified via two examples, and then defined, giving rise to a bicatcgory of concrete monoids over c whose arrows are concrete divisions. Electronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidsoncrete monoids over c.These definitions are illustrated with examples from the theories of semigroups, matrices, vines and automata- With the aid of these definitions, we make functo-rial the well known constructions of the action monoid of an automaton, and the endomorphism monoid of an object of a notion of division for concrete monoidsa category.1 IntroductionIt is always a delicate matter to discuss mathematics informally, but in questions of motivation it often becomes necessary.a notion of division for concrete monoids
Therefore we begin with an informal account of the motivation for this work and ask the reader to bear in mind that when we use expressions such as ’aElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidsural distinction between the property of being abstract or concrete. To illustrate, one might say that an abstract Ml is one in which the elements have no particular interpretation, whereas a concrete set is one in which each element is to be interpreted as something. For example, a set with three e a notion of division for concrete monoidslements is an abstract set, while a set of t hree oranges is a concrete set each of its elements is to be interpreted as a particular orange, real ora notion of division for concrete monoids
imaginary.While it may seem churlish to make this distinction, in the case that these! is the set of elements of a monoid, we give a number of exampleElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidsact sets are different from the natural structural relationships between monoids whose sets of elements an* concrete sets.rhe primary examples of structural relationships between abstract monoids are the embedding ami the quotient. These are both readily (and often) generalized to division: crudely a notion of division for concrete monoidsspeaking, the monoid IỈ divides the monoid .4 if IỈ can be viewed as a submonoid of .4 when one ignores some of the structure of .4. (Formally. li diva notion of division for concrete monoids
ides .4 if there is a submonoid c of .4 with H a quotient of c.) Birkhoff’s Variety Theorem for abstract algebras, Reiterinann's Theorem for finite alElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidsn the structure of the ‘things' which make up the set of elements of a concrete monoid are taken into account, the relationship of division is strengthened. This stronger notion of division for concrete monoids which respects the structure of the elements of the monoid will be called a concrete divi a notion of division for concrete monoidssion and this paper is devoted to finding a general definition for this stronger notion. In particular, we use concrete division to construct a very sa notion of division for concrete monoids
imply defined morphism between concrete monoids which has an associative composition and hence we derive a category of concrete monoids.The purpose ofElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidson will hopefully account for all particular instances of this type of construction.•To convince the reader (particularly the semigroup theorist) that a. good general question when studying concrete monoids is "What are the concrete divisors of this monoid".•To put forth the possibility that, the ca a notion of division for concrete monoidstegory of concrete monoids will be useful ill making functorial various constructions of concrete monoids from other mathematical objects.Example 1.1a notion of division for concrete monoids
(A concrete quotient] Consider the concrete monoid A/ whose elements are braids generated by the elements a and b depicted below.2SolomonThen removingElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoidss a submonoid of B-ị. rhe braid group on 3 strings, the concrete quotient Q of w also gives rise to an obvious concrete division of by Q. Another example of a concrete division in the case that the monoids are transformation monoids will convince the reader of rhe diversity of situations in which co a notion of division for concrete monoidsnstructions of this type arise.Example 1.2 [A concrete division] I his example concerns the case when the elements of the monoid are endofunctions ofa notion of division for concrete monoids
a set. (That is, the monoid is a transformation monoid,)A monoid .4 of transformations of a set A' will be represented as a table each row of the tablElectronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for Conc a notion of division for concrete monoids value of a at i. rhe top row represents the identity element.12222 22 2Electronic Notes in Theoretical Computer Science 12(1998)URL: http://www.elsevier.nl/locate/entcs/volun.el2.html 51 pagesA Notion of Division for ConcGọi ngay
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