1,785 research outputs found

    Permutation of elements in double semigroups

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    Double semigroups have two associative operations ∘,∙\circ, \bullet related by the interchange relation: (a∙b)∘(c∙d)≡(a∘c)∙(b∘d)( a \bullet b ) \circ ( c \bullet d ) \equiv ( a \circ c ) \bullet ( b \circ d ). Kock \cite{Kock2007} (2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer algebra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9.Comment: 24 pages, 11 figures, 4 tables. Final version accepted by Semigroup Forum on 12 March 201

    Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

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    Let XX be a finite set such that ∣X∣=n|X|=n and let i≤j≤ni\leq j \leq n. A group G\leq \sym is said to be (i,j)(i,j)-homogeneous if for every I,J⊆XI,J\subseteq X, such that ∣I∣=i|I|=i and ∣J∣=j|J|=j, there exists g∈Gg\in G such that Ig⊆JIg\subseteq J. (Clearly (i,i)(i,i)-homogeneity is ii-homogeneity in the usual sense.) A group G\leq \sym is said to have the kk-universal transversal property if given any set I⊆XI\subseteq X (with ∣I∣=k|I|=k) and any partition PP of XX into kk blocks, there exists g∈Gg\in G such that IgIg is a section for PP. (That is, the orbit of each kk-subset of XX contains a section for each kk-partition of XX.) In this paper we classify the groups with the kk-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1,k)(k-1,k)-homogeneous groups (for 2<k≤⌊n+12⌋2<k\leq \lfloor \frac{n+1}{2}\rfloor). As a corollary of the classification we prove that a (k−1,k)(k-1,k)-homogeneous group is also (k−2,k−1)(k-2,k-1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the kk-universal transversal property have the (k−1)(k-1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank kk transformation on XX generate a regular semigroup (for 1≤k≤⌊n+12⌋1\leq k\leq \lfloor \frac{n+1}{2}\rfloor). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.Comment: Includes changes suggested by the referee of the Transactions of the AMS. We gratefully thank the referee for an outstanding report that was very helpful. We also thank Peter M. Neumann for the enlightening conversations at the early stages of this investigatio

    Generalized Results on Monoids as Memory

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    We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and prove that they are only as powerful as (finite) valence automata. We observe that certain results proven for monoid automata can be easily lifted to the case of context-free valence grammars.Comment: In Proceedings AFL 2017, arXiv:1708.0622

    Symmetric Groups and Quotient Complexity of Boolean Operations

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    The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory

    The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable

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    We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata and to the decidability of freeness for semigroups generated by two-state invertible-reversible Mealy automata

    Factorization theory: From commutative to noncommutative settings

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    We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω\omega-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×nn \times n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.Comment: 45 page
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