Recent zbMATH articles in MSC 08Ahttps://zbmath.org/atom/cc/08A2022-09-13T20:28:31.338867ZWerkzeugCo-quasiordered residuated systems: an introductionhttps://zbmath.org/1491.030802022-09-13T20:28:31.338867Z"Romano, Daniel A."https://zbmath.org/authors/?q=ai:romano.daniel-abraham\(S\)-acts over a well-ordered monoid with modular congruence latticehttps://zbmath.org/1491.060432022-09-13T20:28:31.338867Z"Stepanova, Alena A."https://zbmath.org/authors/?q=ai:stepanova.alena-andreevnaSummary: This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left \(S\)-act over monoid \(S\) is a set \(A\) upon which \(S\) acts unitarily on the left. In this paper, we consider \(S\)-acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid \(S\) is a linearly ordered set with a minimal element and with a binary operation \(max\), with respect to which \(S\) is obviously a commutative monoid; a well-ordered monoid \(S\) is a well-ordered set with a binary operation \(max\), with respect to which \(S\) is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with \textit{M. S. Kazak} [Sib. Èlektron. Mat. Izv. 16, 1147--1157 (2019; Zbl 1472.06019)], which describes \(S\)-acts over linearly ordered monoids with a linearly ordered congruence lattice and \(S\)-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.An overview of cubic intuitionistic \(\beta\)-subalgebrashttps://zbmath.org/1491.060522022-09-13T20:28:31.338867Z"Muralikrishna, P."https://zbmath.org/authors/?q=ai:muralikrishna.prakasam"Borumand Saeid, A."https://zbmath.org/authors/?q=ai:borumand-saeid.arsham"Vinodkumar, R."https://zbmath.org/authors/?q=ai:vinodkumar.r"Palani, G."https://zbmath.org/authors/?q=ai:palani.g-sThe basic concepts of cubic intuitionistic sets are adopted to \(\beta\)-subalgebras. The obtained results are typical for this theory.
Reviewer: Wiesław A. Dudek (Wrocław)Normally conjugative relationshttps://zbmath.org/1491.080012022-09-13T20:28:31.338867Z"Romano, Daniel A."https://zbmath.org/authors/?q=ai:romano.daniel-abrahamInvertible algebras satisfying associative identities with functional variableshttps://zbmath.org/1491.080022022-09-13T20:28:31.338867Z"Movsisyan, Yu."https://zbmath.org/authors/?q=ai:movsisyan.yuri-m"Gevorgyan, A."https://zbmath.org/authors/?q=ai:gevorgyan.a-h|gevorgyan.albert-r|gevorgyan.a-m|gevorgyan.amirjan-lMeet infinite distributivity for congruence lattices of direct sums of algebrashttps://zbmath.org/1491.080032022-09-13T20:28:31.338867Z"Ghumashyan, Heghine"https://zbmath.org/authors/?q=ai:ghumashyan.heghine"Guričan, Jaroslav"https://zbmath.org/authors/?q=ai:gurican.jaroslavThe authors prove that in a congruence distributive variety the meet infinite distributivity of a congruence lattice is preserved by direct sums of algebras. It means that the congruence lattice of a direct sum of algebras is meet infinitely distributive if and only if a congruence lattice of every constituent algebra is meet infinitely distributive.
Reviewer: Ivan Chajda (Přerov)On some intervals of partial cloneshttps://zbmath.org/1491.080042022-09-13T20:28:31.338867Z"Alekseev, Valeriy B."https://zbmath.org/authors/?q=ai:alekseev.valerii-bSummary: This paper deals with clones, i.e. sets of functions containing all projections and closed under compositions. If \(A\) is any clone from the \(k\)-valued logic \(P_k\), then \(Str(A)\) is the set of all functions from the partial \(k\)-valued logic \(P_k^\ast\), which can be expanded to a function from \(A\). For any clone \(A\) from \(P_k\), the set \(Int(A)\) of all partial clones in \(P_k^\ast\) lying between \(A\) and \(Str(A)\) is investigated. We define a special family \(Z(A)\) of sets of predicates and prove that the lattice of partial clones in \(Int(A)\) (according to inclusion) is isomorphic to the lattice of sets in \(Z(A)\) (according to inclusion). For the set \(J_k\) of all projections in \(P_k\), we prove that the cardinality of \(Int(J_k)\) is continuum. For the set \(Pol_k\) of all polynomials in \(P_k\) where \(k\) is a product of two different prime numbers, we prove that \(Int(Pol_k)\) consists of 7 partial clones which are completely described.All maximal idempotent submonoids of generalized cohypersubstitutions of type \(\tau=(2)\)https://zbmath.org/1491.080052022-09-13T20:28:31.338867Z"Chansuriya, Nagornchat"https://zbmath.org/authors/?q=ai:chansuriya.nagornchatSummary: A generalized cohypersubstitution of type \(\tau\) is a mapping \(\sigma\) which maps every \(n_i\)-ary cooperation symbol \(f_i\) to the coterm \(\sigma(f)\) of type \(\tau= (n_i)_{i\in I}\) . Denote by Cohyp\(_G(\tau)\) the set of all generalized cohypersubstitutions of type \(\tau\). Define the binary operation \(o_{CG}\) on Cohyp\(_G(\tau)\) by \(\sigma_1 \circ_{CG}\sigma_2:=\widehat{\sigma}_1\circ\sigma_2\) for all \(\sigma_1,\sigma_2\in\text{Cohyp}_G(\tau)\) and \(\sigma_{id}(f_i):= f_i\) for all \(i\in I\). Then Cohyp\(_G(\tau):= \{\text{Cohyp}_G(\tau)\), \(\circ_{CG},\sigma_{id}\}\) is a monoid. By \textit{N. Saengsura} and \textit{S. Jermjitpornchai} [``Idempotent and regular cohypersubstitutions of type \(\tau = (2)\)'', Int. J. Pure Appl. Math. 86, No. 4 757--766 (2013); \url{doi:10.12732/ijpam.v86i4.13}], the monoid Cohyp\(_G(2)\) was studied. They characterized and presented the idempotent and regular elements of this monoid. In this present paper, we consider the set of all idempotent elements of the monoid Cohyp\(_G(2)\) and determine all maximal idempotent submonoids of this monoid.Closed sets of finitary functions between products of finite fields of coprime orderhttps://zbmath.org/1491.080062022-09-13T20:28:31.338867Z"Fioravanti, Stefano"https://zbmath.org/authors/?q=ai:fioravanti.stefanoAuthor's abstract: We investigate the finitary functions from a finite product of finite fields \(\prod_{j =1}^m\mathbb{F}_{q_j} = \mathbb{K}\) to a finite product of finite fields \(\prod_{i =1}^n\mathbb{F}_{p_i} = \mathbb{F}\), where \(|\mathbb{K}|\) and \(|\mathbb{F}|\) are coprime. An \((\mathbb{F}, \mathbb{K})\)-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the \(\mathbb{F}_p[\mathbb{K}^\times]\)-submodules of \(\mathbb{F}_p^{\mathbb{K}}\), where \(\mathbb{K}^\times\) is the multiplicative monoid of \(\mathbb{K}= \prod_{i=1}^m \mathbb{F}_{q_i} \). Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct \((\mathbb{F}, \mathbb{K})\)-linearly closed clonoids.
Reviewer: Wiesław A. Dudek (Wrocław)Full formulas induced by full termshttps://zbmath.org/1491.080072022-09-13T20:28:31.338867Z"Kumduang, Thodsaporn"https://zbmath.org/authors/?q=ai:kumduang.thodsaporn"Leeratanavalee, Sorasak"https://zbmath.org/authors/?q=ai:leeratanavalee.sorasakSummary: The algebraic systems is a well-established structure of classical universal algebra. An algebraic system is a triple consisting a nonempty set together with the sequence of operation symbols and the sequence of relation symbols. To express the primary properties of algebraic sysytems one needs the notion of formulas. The paper is devoted to studying of the structures related to full formulas which are extensional concepts construced from full terms. Defining a superposition operation on the set of full formulas one obtains a many-sorted algebra which satisfies the superassociative law. In particular, we introduce a natural concept of a full hypersubstitution for algebraic systems which extends the concept of full hypersubstitutions of algebras, i.e., the mappings which send operation symbols to full terms of the same arities and relation symbols to full formulas of the corresponding arities. Together with one associative operation on the collection of full hypersubstitutions for algebraic systems, we obtain a semigroup of full hypersubstitutions for algebraic systems.C-maximal strong partial clones and the inclusion structure of Boolean weak baseshttps://zbmath.org/1491.080082022-09-13T20:28:31.338867Z"Lagerkvist, Victor"https://zbmath.org/authors/?q=ai:lagerkvist.victor"Roy, Biman"https://zbmath.org/authors/?q=ai:roy.bimanSummary: Strong partial clones are composition closed sets of partial operations containing all partial projections, characterizable as partial polymorphisms of sets of relations \(\Gamma\) (pPol\((\Gamma))\). If \textsf{C} is a clone it is known that the set of all strong partial clones whose total component equals \textsf{C}, has a greatest element pPol\((\Gamma_\omega)\), where \(\Gamma_\omega\) is called a weak base. Weak bases have seen applications in computer science due to their usefulness for proving complexity classifications for constraint satisfaction related problems. In this article we (1) completely describe the inclusion structure between pPol\((\Gamma_\omega)\), pPol\((\Delta_\omega)\) for all Boolean weak bases \(\Gamma_\omega\) and \(\Delta_\omega\) and (2) in many such cases prove that the strong partial clones in question uniquely cover each other.\(S(\bar{n_i},Y_i)\)-terms and their algebraic propertieshttps://zbmath.org/1491.080092022-09-13T20:28:31.338867Z"Phuapong, Sarawut"https://zbmath.org/authors/?q=ai:phuapong.sarawut"Pookpienlert, Chollawat"https://zbmath.org/authors/?q=ai:pookpienlert.chollawatThe authors interoduce a special kind of terms (the name is in the title) and prove that the set of all these terms for a given similarity type forms a Menger algebra. They study algebraic properties of this Menger algbera and hypersubstitutions.
Reviewer: Ivan Chajda (Přerov)The Menger algebra of terms induced by order-decreasing transformationshttps://zbmath.org/1491.080102022-09-13T20:28:31.338867Z"Wattanatripop, Khwancheewa"https://zbmath.org/authors/?q=ai:wattanatripop.khwancheewa"Changphas, Thawhat"https://zbmath.org/authors/?q=ai:changphas.thawhatThe authors define the so-called order-decreasing terms of a given similarity type by induction. They prove that the set of these terms is closed under superposition and hence forms a Menger algebra. They continue with hypersubstitutions of order-decreasing terms and related so-called full closed identities and with order-decreasing full closed varieties.
Reviewer: Ivan Chajda (Přerov)Group congruences on trioidshttps://zbmath.org/1491.080112022-09-13T20:28:31.338867Z"Zhuchok, A. V."https://zbmath.org/authors/?q=ai:zhuchok.anatolii-vSummary: We describe all group congruences on an arbitrary trioid and present the least group congruence on a trioid with an inverse semigroup, the least group congruence on a trioid with an orthodox semigroup and the least group congruence on a trioid with a regular semigroup.Hardness results for the subpower membership problemhttps://zbmath.org/1491.080122022-09-13T20:28:31.338867Z"Shriner, Jeff"https://zbmath.org/authors/?q=ai:shriner.jeffStructure of relatively free trioidshttps://zbmath.org/1491.080132022-09-13T20:28:31.338867Z"Zhuchok, A. V."https://zbmath.org/authors/?q=ai:zhuchok.anatolii-vSummary: \textit{J.-L. Loday} and \textit{M. Ronco} [Contemp. Math. 346, 369--398 (2004; Zbl 1065.18007)] introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank \(1\) and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free \(n\)-nilpotent trioid, the free left (right) \(n\)-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.Semigroups of an inductive composition of termshttps://zbmath.org/1491.201122022-09-13T20:28:31.338867Z"Kitpratyakul, Pongsakorn"https://zbmath.org/authors/?q=ai:kitpratyakul.pongsakorn"Pibaljommee, Bundit"https://zbmath.org/authors/?q=ai:pibaljommee.bunditSemigroups of linear tree languageshttps://zbmath.org/1491.201402022-09-13T20:28:31.338867Z"Kitpratyakul, Pongsakorn"https://zbmath.org/authors/?q=ai:kitpratyakul.pongsakorn"Pibaljommee, Bundit"https://zbmath.org/authors/?q=ai:pibaljommee.bunditOn some associative formula with functional variableshttps://zbmath.org/1491.201462022-09-13T20:28:31.338867Z"Harutyunyan, D. N."https://zbmath.org/authors/?q=ai:harutyunyan.d-nSummary: In work of \textit{Yu. Movsisyan} and \textit{A. Gevorgyan} [Asian-Eur. J. Math. 14, No. 1, Article ID 2050155, 16 p. (2021; Zbl 1491.08002)], it was proved that the reversible algebra \((Q,\Sigma)\) satisfying the following second-order formula
\[\forall X,Y,\exists X',Y'\forall x,y,z(X(Y'(x,y),z)=Y(x,X'(y,z))),\]
is linear over the group. In the current work a more general result is proved that the regular and division algebra \((Q,\Sigma)\) with the mentioned second-order formula is endolinear over the group.Constructions of polynomially complete quasigroups of arbitrary orderhttps://zbmath.org/1491.201482022-09-13T20:28:31.338867Z"Artamonov, V. A."https://zbmath.org/authors/?q=ai:artamonov.vyacheslavovich-aleksandrovich|artamonov.vyacheslav-a"Chakrabarti, S."https://zbmath.org/authors/?q=ai:chakrabarti.subhadip|chakrabarti.sucharita|chakrabarti.sayak|chakrabarti.subhroneel|chakrabarti.soumen|chakrabarti.sukanya|chakrabarti.sayan-k|chakrabarti.satyabrata|chakrabarti.subrata-k|chakrabarti.sanjib-kumar|chakrabarti.subir-k|chakrabarti.sandip-k|chakrabarti.sadasiv|chakrabarti.saikat|chakrabarti.sulagna|chakrabarti.saswat|chakrabarti.sucheta|chakrabarti.soumya|chakrabarti.satish-chandra"Markov, V. T."https://zbmath.org/authors/?q=ai:markov.viktor-timofeevich"Pal, S. K."https://zbmath.org/authors/?q=ai:pal.swapan-kr|pal.surjya-k|pal.sayan-kumar|pal.surya-kant|pal.saibal-kumar|pal.sandip-kumar|pal.sudipta-kumar|pal.shiv-kumar|pal.sudip-kumar|pal.sankar-kr|pal.sankar-kumarMal'cev classes of left quasigroups and quandleshttps://zbmath.org/1491.201502022-09-13T20:28:31.338867Z"Bonatto, Marco"https://zbmath.org/authors/?q=ai:bonatto.marco"Fioravanti, Stefano"https://zbmath.org/authors/?q=ai:fioravanti.stefanoSummary: In this paper we investigate some Mal'cev classes of varieties of left quasigroups. We prove that the weakest non-trivial Mal'cev condition for a variety of left quasigroups is having a Mal'cev term and that every congruence meet-semidistributive variety of left quasigroups is congruence arithmetic. Then we specialize to the setting of quandles for which we prove that the congruence distributive varieties are those which have no non-trivial finite models.Every Moufang loop of odd order has nontrivial nucleushttps://zbmath.org/1491.201512022-09-13T20:28:31.338867Z"Csörgő, Piroska"https://zbmath.org/authors/?q=ai:csorgo.piroskaIn this paper, the author give a negative answer by proving that every Moufang loop of odd order has nontrivial nucleus. To obtain his result the author focus on Moufang loops satisfying any one of the three equivalent Moufang identities: $((xy)x)z = x(y(xz))$; $((xy)z)y = x(y(zy))$; $(xy)(zx) = x(yz)x$. A class of Moufang loop can be viewed as a ``group with weakened associativity''. The links between algebra, geometry, and group theory explain the importance of this class. Thus, according to the author, the main objective of the paper is to solve one of the main problems in the area of Moufang loops, namely: does there exist a Moufang loop of odd order with trivial nucleus? So, the author gives a negative answer. For the proof of the main result, the author used the structural properties of the multiplication group of Moufang loops of odd order. The author's proof is completely group theoretical relying on the theory of connected transversals. This concept was introduced by \textit{M. Niemenmaa} and \textit{T. Kepka} [J. Algebra 135, No. 1, 112--122 (1990; Zbl 0706.20046)]. Using their characterization theorem, the author transforms loop theoretical problems into group theoretical problems.
Reviewer: C. Pereira da Silva (Curitiba)Semi-lattice of varieties of quasigroups with linearityhttps://zbmath.org/1491.201612022-09-13T20:28:31.338867Z"Sokhatsky, F. M."https://zbmath.org/authors/?q=ai:sokhatsky.fedir-m"Krainichuk, H. V."https://zbmath.org/authors/?q=ai:krainichuk.halyna-v"Sydoruk, V. A."https://zbmath.org/authors/?q=ai:sydoruk.v-aSummary: A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a \textit{parastrophic orbit} or a \textit{truss}. A parastrophically closed semi-lattice of classes is a \textit{bunch}.
A \textit{linearity bunch} is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of \textit{middle linearity} and \textit{middle alinearity} are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented.Partial Menger algebras of termshttps://zbmath.org/1491.201662022-09-13T20:28:31.338867Z"Denecke, K."https://zbmath.org/authors/?q=ai:denecke.klaus"Hounnon, H."https://zbmath.org/authors/?q=ai:hounnon.hippolyteMenger algebras of \(k\)-commutative \(n\)-place functionshttps://zbmath.org/1491.201682022-09-13T20:28:31.338867Z"Dudek, Wieslaw A."https://zbmath.org/authors/?q=ai:dudek.wieslaw-aleksander"Trokhimenko, Valentin S."https://zbmath.org/authors/?q=ai:trokhimenko.valentin-sLet \({g_1},\dots ,{g_n} = g_1^n\), \(g,\dots ,g = \mathop g\limits^n \). If \({\mathcal T}({A^n},A)\) is the set of all \(n\)-place everywhere defined functions on a nonempty set \(A\) and the \(n + 1\)-ary Menger superposition operation is defined by \({\mathcal O}(f,g_{_1}^n)(x_1^n) = f(g_1^n(x_1^n))\) then every subset \(\Phi \subseteq {\mathcal T}({A^n},A)\) closed for \({\mathcal O}\) is a Menger algebra \((\Phi ,{\mathcal O})\) of rank \(n\); alternative notation for Menger superposition is \({\mathcal O}(f,g_1^n) = f[g_1^n]\). An \(n\)-place function \(f \in {\mathcal T}({A^n},A)\) is \(k\)-commutative, \(1 \leqslant k \leqslant n - 1\), if for all \(a,b \in A\) and all \(i = 1,2,\dots ,n - k\) holds\(f(\mathop a\limits^{i - 1} ,\mathop b\limits^k ,\mathop a\limits^{n - i - k + 1} ) = f(\mathop b\limits^{i - 1} ,\mathop a\limits^k ,\mathop b\limits^{n - i - k + 1} )\); \(\mathfrak{M}_n^k(A)\) is the set of all \(k\)-commutative \(n\)-place functions on the set \(A\), \((\mathfrak{M}_n^k(A),{\mathcal O})\) is a subalgebra of \(({\mathcal T}({A^n},A),{\mathcal O})\). It is shown that a Menger algebra \((G,o)\) of rank \(n \geqslant 2\) can be isomorphically embedded into a Menger algebra \((\mathfrak{M}_n^k(A),{\mathcal O})\) iff \(x[\mathop y\limits^{i - 1} \mathop z\limits^k \mathop y\limits^{n - i - k + 1} ] = x[\mathop z\limits^{i - 1} \mathop y\limits^k \mathop z\limits^{n - i - k + 1} ]\).
A subset \(H\) of a Menger algebra \((G,o)\) is
\begin{itemize}
\item[(1)] an \(s\)-ideal iff \((\forall h,x_1^n \in G)(h \in H \Rightarrow h[x_1^n] \in H)\);
\item[(2)] a \(v\)-ideal iff \((\forall x,h_1^n \in G)(h_1^n \in H \Rightarrow x[h_1^n] \in H)\);
\item[(3)] an \(sv\)-ideal iff it is both an \(s\)- and a \(v\)-ideal.
\end{itemize}
Denote by \({C_A} = \{ {f_a}:a \in A,(\forall {a_1},\dots ,{a_n} \in A)f(a_1^n) = a\} \) the set of all constant functions. It is shown that \({C_A}\) is densely embedded in \((\mathfrak{M}_n^k(A),{\mathcal O})\) (every nontrivial congruence on parent induces a nontrivial congruence on \({C_A}\)). The set of constants and dense embeddings are used to present a description of all Menger algebras isomorphic to some \((\mathfrak{M}_n^k(A),{\mathcal O})\) and to prove, that all automorphisms of \((\mathfrak{M}_n^k(A),{\mathcal O})\) are inner.
Reviewer: Jaak Henno (Tallinn)Left translations and isomorphism theorems for Menger algebras of rank \(n\)https://zbmath.org/1491.201692022-09-13T20:28:31.338867Z"Kumduang, Thodsaporn"https://zbmath.org/authors/?q=ai:kumduang.thodsaporn"Leeratanavalee, Sorasak"https://zbmath.org/authors/?q=ai:leeratanavalee.sorasakSummary: Let \(n\) be a fixed natural number. Menger algebras of rank \(n\) can be regarded as a canonical generalization of arbitrary semigroups. This paper is concerned with studying algebraic properties of Menger algebras of rank \(n\) by first defining a special class of full \(n\)-place functions, the so-called a left translation, which possess necessary and sufficient conditions for an \((n+1)\)-groupoid to be a Menger algebra of rank \(n\). The isomorphism parts begin with introducing the concept of homomorphisms, and congruences in Menger algebras of rank \(n\). These lead us to establish a quotient structure consisting a nonempty set factored by such congruences together with an operation defined on its equivalence classes. Finally, the fundamental homomorphism theorem and isomorphism theorems for Menger algebras of rank \(n\) are given. As a consequence, our results are significant in the study of algebraic theoretical Menger algebras of rank \(n\). Furthermore, we extend the usual notions of ordinary semigroups in a natural way.Partial associativity and rough approximate groupshttps://zbmath.org/1491.201862022-09-13T20:28:31.338867Z"Gowers, W. T."https://zbmath.org/authors/?q=ai:gowers.william-timothy"Long, J."https://zbmath.org/authors/?q=ai:long.jianyong|long.jie|long.jianwu|long.jeff|long.jianceng|long.jinling|long.junyun|long.jingfan|long.jianmin|long.jiang|long.jianzhong|long.jason|long.jinghua|long.jiangqi|long.jiancheng|long.jun|long.junsheng|long.junbo|long.jiaping|long.jancis|long.jianyu|long.junyan|long.jing|long.jianhui|long.jane|long.jianjun|long.jihaoSummary: Suppose that a binary operation \(\circ\) on a finite set \(X\) is injective in each variable separately and also associative. It is easy to prove that \((X, \circ)\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x, y, z) \in X^3\) satisfy the equation \(x \circ (y \circ z) = (x \circ y) \circ z\). Other results in additive combinatorics would lead one to expect that there must be an underlying `group-like' structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.A dichotomy theorem for the inverse satisfiability problemhttps://zbmath.org/1491.680892022-09-13T20:28:31.338867Z"Lagerkvist, Victor"https://zbmath.org/authors/?q=ai:lagerkvist.victor"Roy, Biman"https://zbmath.org/authors/?q=ai:roy.bimanSummary: The inverse satisfiability problem over a set of Boolean relations \(\Gamma\) (\(\operatorname{Inv-SAT}(\Gamma)\)) is the computational decision problem of, given a relation \(R\), deciding whether there exists a \(\operatorname{SAT}(\Gamma)\) instance with \(R\) as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite \(\Gamma\) containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that \(\operatorname{Inv-SAT}(\Gamma)\) is always either tractable or co-NP-complete for all finite sets of relations \(\Gamma\), thus solving a problem open since 1998. Very few of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when \(\Gamma\) is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite \(\Gamma\) such that \(\operatorname{Inv-SAT}(\Gamma)\) is tractable even though there exists finite \(\Delta\subset\Gamma\) such that \(\operatorname{Inv-SAT}(\Delta)\) is co-NP-complete.
For the entire collection see [Zbl 1388.68010].A proof of the CSP dichotomy conjecturehttps://zbmath.org/1491.681282022-09-13T20:28:31.338867Z"Zhuk, Dmitriy"https://zbmath.org/authors/?q=ai:zhuk.dmitriy-n