Finite degree clones are undecidable

Moore, Matthew

doi:10.1016/j.tcs.2019.09.014

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…Conversely, Barto proved Valeriote's conjecture [4]: Every finite, finitely related algebra which generates a congruence modular variety has few subpowers. In [14] Moore showed that the question whether a given finite algebra is finitely related is undecidable. As a result, characterizations based on finite relatedness gain greater significance.…”

Section: Introductionmentioning

confidence: 99%

Not all nilpotent monoids are finitely related

Steindl

2024

Algebra Univers.

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A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

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Section: Introductionmentioning

confidence: 99%

Not all nilpotent monoids are finitely related

Steindl

2024

Algebra Univers.

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“…Moore [13] showed that the question whether a given finite algebra is finitely related is undecidable. This makes characterizations by finite relatedness even more valuable.…”

Section: Introductionmentioning

confidence: 99%

Not all nilpotent monoids are finitely related

Steindl¹

2022

Preprint

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A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. This is the first known example where adjoining an identity to a finitely related semigroup yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

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Submaximal clones over a three-element set up to minor-equivalence

Vucaj,

Zhuk

2024

Algebra Univers.

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We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form $$f(x_1,\dots ,x_n)\approx g(y_1,\dots ,y_m)$$ f ( x 1 , ⋯ , x n ) ≈ g ( y 1 , ⋯ , y m ) , also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $${\text {CSP}}$$ CSP of a finite structure $$\mathbb {A}$$ A only depends on the set of minor identities satisfied by the polymorphism clone of $$\mathbb {A}$$ A . In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write $$\mathcal {C}\ {\preceq _{\textrm{m}}}\ \mathcal {D}$$ C ⪯ m D if there exists a minor homomorphism from $$\mathcal {C}$$ C to $$\mathcal {D}$$ D . We show that the aforementioned poset has only three submaximal elements.

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Finite degree clones are undecidable

Cited by 3 publications

References 33 publications

Not all nilpotent monoids are finitely related

Not all nilpotent monoids are finitely related

Not all nilpotent monoids are finitely related

Submaximal clones over a three-element set up to minor-equivalence

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