Lesson plan
Automatic semigroup
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator. Formally, let S be a semigroup and A be a finite set of generators. Then an automatic structure for S with respect to A consists of a regular language L over A such that every element of S has at least one representative in L and such that for each a \in A \cup \{\varepsilon\} , the relation consisting of pairs (u,v) with ua = v is regular, viewed as a subset of (A # × A # )*. Here A # is A augmented with a padding symbol. The concept of an automatic semigroup was generalized from automatic groups by Campbell et al. (2001) Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).
Article summary
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator. Formally, let S be a semigroup and A be a finite set of generators. Then an automatic structure for S with respect to A consists of a regular language L over A such that every element of S has at least one representative in L and such that for each a \in A \cup \{\varepsilon\} , the relation consisting of pairs (u,v) with ua = v is regular, viewed as a subset of (A # × A # )*. Here A # is A augmented with a padding symbol. The concept of an automatic semigroup was generalized from automatic groups by Campbell et al. (2001) Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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