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Green index in semigroups: generators, presentations and automatics structures
"... Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. This gives rise to a partition of the complement S \ T and to each equivalence class of this partition we naturally associate a relative Schützenberger group. We show how generating sets fo ..."
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Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. This gives rise to a partition of the complement S \ T and to each equivalence class of this partition we naturally associate a relative Schützenberger group. We show how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity, finite presentability (for extensions) and finite Malcev presentability (in the case of groupembeddable semigroups). These results provide common generalisations of several classical results from group theory and Rees index results from semigroup theory.
Tutorial  Computing with Semigroups in GAP
"... emigroups, and Rees matrix semigroups; 3. The power set semigroup will be of interest to potential GAP developers, showing how to create new types of multiplicative elements, and how to use finitely presented semigroups, semigroup homomorphisms and the built in ToddCoxeter enumerator; 4. Endomorp ..."
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emigroups, and Rees matrix semigroups; 3. The power set semigroup will be of interest to potential GAP developers, showing how to create new types of multiplicative elements, and how to use finitely presented semigroups, semigroup homomorphisms and the built in ToddCoxeter enumerator; 4. Endomorphisms of the symmetric group deals with selecting element representation for efficiency, moving between semigroups using isomorphisms, and studying semigroups whose elements are themselves homomorphisms; 5. The Heisenberg group showcases GAP's features for working with infinite finitely presented semigroups, such as the KnuthBendix procedure. 1 Endomorphisms of a finite chain In this section we use GAP to investigate the structure of O n , the semigroup of endomorphisms of a finite chain. The user will gain experience in working with: ffl transformation semigroups; ffl congruences and Rees congruences; ffl Green's relations, Green's classes and eggboxes. Consider the set [n] = f1; 2;
GROUPS THAT TOGETHER WITH ANY TRANSFORMATION GENERATE REGULAR SEMIGROUPS OR IDEMPOTENT GENERATED SEMIGROUPS
"... Abstract. Let a be a noninvertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈 G, a 〉 \ G is a subsemigroup, consisting of all noninvertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elem ..."
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Abstract. Let a be a noninvertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈 G, a 〉 \ G is a subsemigroup, consisting of all noninvertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted 〈a g  g ∈ G〉. We classify the finite permutation groups G on a finite set X such that the semigroups 〈G, a〉, 〈G, a〉\G, and 〈a g  g ∈ G 〉 are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups 〈 G, a 〉 \ G and 〈 a g  g ∈ G 〉 are generated by their idempotents for all noninvertible transformations of X.