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- Free_monoid abstract "In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the monoid operation is string concatenation. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string. It is usually denoted A+.More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set.As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.".
- Free_monoid wikiPageID "523166".
- Free_monoid wikiPageRevisionID "582427998".
- Free_monoid date "August 2013".
- Free_monoid hasPhotoCollection Free_monoid.
- Free_monoid reason "At least, a reference to a definition of 'homomorphic pre-image' and a reference to a proof should be added here. Also, rephrasing the sentence in singular may improve clarity.".
- Free_monoid reason "Define 'primitive substitution' and 'balanced'; indicate whether the notion of a 'sturmian substitution' is subsumed by the definition of a 'sturmian endomorphism' or whether is something different.".
- Free_monoid reason "Give a definition of 'graded monoid' or a reference to it. A Wikipedia article 'graded ring' exists, but it doesn't define graded monoids. In case 'graded' is meant just informally, the sentence should better be rephrased as e.g. 'Due to this additional functionality, some authors call a free monoid a ⟨graded⟩ one.'".
- Free_monoid reason "Please give a definition of 'factors through' or a link to it.".
- Free_monoid reason "Somebody please cross-check my example: Aren't generator sets restricted to single-character sets, so my generator set wouldn't be valid? But I can't imagine another one. On the other hand, each member of N can obviously be decomposed into a sequence of my generators in a unique way, e.g. 1010101 → 101⋅0⋅101".
- Free_monoid reason "The image of f is a set of strings, while w° is a string. None of the meanings of 'is contained in' fits, i.e. neither 'is an element of', nor 'is a subset of', nor 'is a substring of'. Probably, every value f is required to be a substring of w°.".
- Free_monoid reason "Up to here, string projection has been defined only for single letters, not for arbitrary strings. However, the empty string ε is not a letter.".
- Free_monoid reason "What is the difference between a set of generators and a set of free generators? Up to here, only the latter has been defined.".
- Free_monoid subject Category:Combinatorics_on_words.
- Free_monoid subject Category:Formal_languages.
- Free_monoid subject Category:Free_algebraic_structures.
- Free_monoid subject Category:Semigroup_theory.
- Free_monoid type Artifact100021939.
- Free_monoid type FreeAlgebraicStructures.
- Free_monoid type Object100002684.
- Free_monoid type PhysicalEntity100001930.
- Free_monoid type Structure104341686.
- Free_monoid type Whole100003553.
- Free_monoid type YagoGeoEntity.
- Free_monoid type YagoPermanentlyLocatedEntity.
- Free_monoid comment "In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the monoid operation is string concatenation. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string.".
- Free_monoid label "Free monoid".
- Free_monoid label "Monoide livre".
- Free_monoid label "自由么半群".
- Free_monoid sameAs Monoide_livre.
- Free_monoid sameAs m.02l8lg.
- Free_monoid sameAs Q5500266.
- Free_monoid sameAs Q5500266.
- Free_monoid sameAs Free_monoid.
- Free_monoid wasDerivedFrom Free_monoid?oldid=582427998.
- Free_monoid isPrimaryTopicOf Free_monoid.