109 research outputs found

    SS-preclones and the Galois connection SPol{}^S\mathrm{Pol}-SInv{}^S\mathrm{Inv}, Part I

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    We consider SS-operations f ⁣:AnAf \colon A^{n} \to A in which each argument is assigned a signum sSs \in S representing a "property" such as being order-preserving or order-reversing with respect to a fixed partial order on AA. The set SS of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of SS-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all SS-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of SS-preclone. We introduce SS-relations ϱ=(ϱs)sS\varrho = (\varrho_{s})_{s \in S}, SS-relational clones, and a preservation property (fSϱf \mathrel{\stackrel{S}{\triangleright}} \varrho), and we consider the induced Galois connection SPol{}^S\mathrm{Pol}-SInv{}^S\mathrm{Inv}. The SS-preclones and SS-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all SS-preclones on AA.Comment: 31 page

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    On when the union of two algebraic sets is algebraic

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    In universal algebraic geometry, an algebra is called an equational domain if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.Comment: 50 pages, 1 figure, 1 tabl

    Clones of pigmented words and realizations of special classes of monoids

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    Clones are generalizations of operads forming powerful instruments to describe varieties of algebras wherein repeating variables are allowed in their relations. They allow us in this way to realize and study a large range of algebraic structures. A functorial construction from the category of monoids to the category of clones is introduced. The obtained clones involve words on positive integers where letters are pigmented by elements of a monoid. By considering quotients of these structures, we construct a complete hierarchy of clones involving some families of combinatorial objects. This provides clone realizations of some known and some new special classes of monoids as among others the variety of left-regular bands, bounded semilattices, and regular band monoids.Comment: 41 page

    On constructing topology from algebra

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    In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on the set N. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the following semigroups: the monoid of all injective functions on N, the monoid of continuous transformations of the Hilbert cube [0, 1]N, the monoid of continuous transformations of the Cantor space 2N, and the monoid of endomorphisms of the countably infinite atomless boolean algebra. With the exception of the continuous transformation monoid of the Hilbert cube, we also show that all of the above semigroups admit a second countable semigroup topology such that every semigroup homomorphism from the semigroup to a second countable topological semigroup is continuous. In a recent paper, Bleak, Cameron, Maissel, Navas, and Olukoya use a theorem of Rubin to describe the automorphism groups of the Higman-Thompson groups Gₙ,ᵣ via their canonical Rubin action on the Cantor space. In particular they embed these automorphism groups into the rational group R of transducers introduced by Grigorchuk, Nekrashevich, and Sushchanskii. We generalise these transducers to be more suitable to higher dimensional Cantor spaces and give a similar description of the automorphism groups of the Brin-Thompson groups Vₙ (although we do not give an embedding into R). Using our description, we show that the outer automorphism group Out(V₂) of V₂ is isomorphic to the wreath product of Out(1V₂) with the symmetric group on points

    Foundations of Software Science and Computation Structures

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    This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
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