Relatively inherently nonfinitely q-based semigroups

Jackson, Marcel; Volkov, M. V.

doi:10.1090/s0002-9947-08-04798-3

Cited by 14 publications

(20 citation statements)

References 25 publications

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“…This will complete the proof of Theorem 9.1. (The construction has been adapted by Jackson and Volkov [27] to show that "almost all" finite semigroups are not finitely q-based. )…”

Section: Cyclic Semigroupsmentioning

confidence: 99%

The axiomatizability of topological prevarieties

Clark

Davey

Jackson

et al. 2008

Advances in Mathematics

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We investigate first-order axiomatic descriptions of naturally occurring classes of Boolean topological structures (these structures can have operations and relations, and carry a compatible compact Hausdorff topology with a basis of clopen sets). Our methods utilize inverse limits and ultraproducts of finite structures. We illustrate the range of possible axiomatizations of these classes with applications of our methods to Boolean topological lattices, graphs, ordered structures, unary algebras and semigroups. For example, whereas the class of all k-colorable graphs is known to be axiomatizable by universal Horn sentences, we find the class of continuously k-colorable Boolean topological graphs is not even first-order axiomatizable.

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“…This will complete the proof of Theorem 9.1. (The construction has been adapted by Jackson and Volkov [27] to show that "almost all" finite semigroups are not finitely q-based. )…”

Section: Cyclic Semigroupsmentioning

confidence: 99%

The axiomatizability of topological prevarieties

Clark

Davey

Jackson

et al. 2008

Advances in Mathematics

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“…We now add to this list of complex behaviour by showing that they are inherently nondualisable. We mention that the template T originated in the proof of early versions of the following theorem, using the notion of a "homotopy", in the style of [17].…”

Section: Nilpotent and Monogenic Semigroupsmentioning

confidence: 99%

Natural dualities, nilpotence and projective planes

Jackson

2015

Algebra Univers.

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We use an interpretation of projective planes to show the inherent nondualisability of some finite semigroups. The method is sufficiently flexible to demonstrate the nondualisability of (asymptotically) almost all finite semigroups as well as to give a fresh proof of the Quackenbush-Szabó result that any finite group with a nonabelian Sylow subgroup is nondualisable. A novel feature is that the ostensibly different notions of nilpotence for semigroups, nilpotence for groups, and the property of being nonorthodox for a completely 0-simple semigroup are unified by way of a single construction. We also give a semigroup example of two dualisable finite semigroups whose direct product is inherently nondualisable.2000 Mathematics Subject Classification. Primary: 08C20. Secondary: 20M07, 20M17.

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“…The most general results about finite semigroups with respect to finite bases of quasi-identities appeared in [16]. We recall some definitions.…”

Section: Historical Backgroundmentioning

confidence: 99%

Semigroups embeddable in hyperplane face monoids

2013

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The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.

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Relatively inherently nonfinitely q-based semigroups

Cited by 14 publications

References 25 publications

The axiomatizability of topological prevarieties

The axiomatizability of topological prevarieties

Natural dualities, nilpotence and projective planes

Semigroups embeddable in hyperplane face monoids

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