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Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, every set of generators for a free monoid or semigroup S contains the free generators, since a free generator has word length 1 and hence can only be generated by itself. It follows that a free semigroup or monoid is finitely generated if and only if ...
The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory . As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system).
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
Let denote the free monoid on a set of generators , that is, the set of all strings written in the alphabet .The asterisk is a standard notation for the Kleene star.An independency relation on the alphabet then induces a symmetric binary relation on the set of strings : two strings , are related, , if and only if there exist ,, and a pair (,) such that = and =.
A science project is an educational activity for students involving experiments or construction of models in one of the science disciplines. Students may present their science project at a science fair, so they may also call it a science fair project. Science projects may be classified into four main types.
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums , where for each and r g = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G.
For every category C, the free strict monoidal category Σ(C) can be constructed as follows: its objects are lists (finite sequences) A 1, ..., A n of objects of C; there are arrows between two objects A 1, ..., A m and B 1, ..., B n only if m = n, and then the arrows are lists (finite sequences) of arrows f 1: A 1 → B 1, ..., f n: A n → B ...
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