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  • 1
    Electronic Resource
    Electronic Resource
    Presentations for Two Extensions of the Monoid of Order-Preserving Mappings on a Finite Chain (2000)
    Springer
    Southeast Asian bulletin of mathematics 24 (2000), S. 1-7 
    Details
    ISSN: 0219-175X
    Keywords: presentations ; monoids ; transformations ; orientation-preserving ; order-preserving
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to give presentations for the monoids of orientation-preserving mappings on a finite chain of order n, and orientation-preserving or reversing mappings on such a chain. Both these monoids are natural extensions of the monoid of order-preserving mappings. The obtained presentations are on two and three generators, respectively, and have n + 2 and n + 6 relations, respectively.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10012-000-0001-1
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  • 2
    Electronic Resource
    Electronic Resource
    Log-linear varieties of semigroups (1995)
    Springer
    Algebra universalis 33 (1995), S. 370-374 
    Details
    ISSN: 1420-8911
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A necessary and sufficient condition for a variety of semigroups to be log-linear is found in terms of identities. In particular, every log-linear variety of semigroups is hereditarily finitely based. Also, it can be effectively decided whether a finite semigroup generates a log-linear variety.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01190705
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  • 3
    Electronic Resource
    Electronic Resource
    Minimal Presentations and Efficiency of Semigroups (2000)
    Springer
    Semigroup forum 60 (2000), S. 231-242 
    Details
    ISSN: 1432-2137
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H 2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D 2 n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002339910016
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  • 4
    Electronic Resource
    Electronic Resource
    Groups and actions in transformation semigroups (1998)
    Springer
    Mathematische Zeitschrift 228 (1998), S. 435-450 
    Details
    ISSN: 0025-5874
    Keywords: Mathematics Subject Classification (1991):20M20, 20B40, 20M10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. Let $S$ be a transformation semigroup of degree $n$ . To each element $s\in S$ we associate a permutation group $G_R(s)$ acting on the image of $s$ , and we find a natural generating set for this group. It turns out that the $\mathcal{R}$ -class of $s$ is a disjoint union of certain sets, each having size equal to the size of $G_R(s)$ . As a consequence, we show that two $\mathcal{R}$ -classes containing elements with equal images have the same size, even if they do not belong to the same $\mathcal{D}$ -class. By a certain duality process we associate to $s$ another permutation group $G_L(s)$ on the image of $s$ , and prove analogous results for the $\mathcal{L}$ -class of $S$ . Finally we prove that the Schützenberger group of the $\mathcal{H}$ -class of $s$ is isomorphic to the intersection of $G_R(s)$ and $G_L(s)$ . The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00004628
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  • 5
    Electronic Resource
    Electronic Resource
    On Finite Presentability of Direct Products of Semigroups (2000)
    Springer
    Algebra colloquium 7 (2000), S. 83-91 
    Details
    ISSN: 0219-1733
    Keywords: semigroup ; generators ; presentation ; direct product ; connected graph
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A finite semigroup S is said to preserve finite generation (resp., presentability) in direct products, provided that, for every infinite semigroup T, the direct product S × T is finitely generated (resp., finitely presented) if and only if T is finitely generated (resp., finitely presented). The main result of this paper is a constructive necessary and sufficient condition for S to preserve both finite generation and presentability in direct products. The condition is that certain graphs, (s), one for each s ɛ S, are all connected. The main result is illustrated in three examples, one of which exhibits a 4-element semigroup that preserves finite generation but not finite presentability in direct products.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10011-000-0083-4
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