Recent zbMATH articles in MSC 08https://zbmath.org/atom/cc/082022-11-17T18:59:28.764376ZWerkzeugProving and rewritinghttps://zbmath.org/1496.030452022-11-17T18:59:28.764376Z"Goguen, Joseph A."https://zbmath.org/authors/?q=ai:goguen.joseph-amadeeSummary: This paper presents some ways to prove theorems in first and second order logic, such that rewriting does the routine work automatically, and partially successful proofs often return information that suggests what to try next. The theoretical framework makes extensive use of general algebra, and main results include an extension of many-sorted equational logic to universal quantification over functions, some techniques for handling first order logic, and some structural induction principles. The OBJ language is used for illustration, and initiality is a recurrent theme.
For the entire collection see [Zbl 0763.68011].Regular double \(p\)-algebrashttps://zbmath.org/1496.060122022-11-17T18:59:28.764376Z"Adams, M. E."https://zbmath.org/authors/?q=ai:adams.mike-e|adams.michael-e"Sankappanavar, Hanamantagouda P."https://zbmath.org/authors/?q=ai:sankappanavar.hanamantagouda-p"Vaz de Carvalho, Júlia"https://zbmath.org/authors/?q=ai:vaz-de-carvalho.julia(no abstract)Solvability of a triple of binary relations with an application to Green's relationshttps://zbmath.org/1496.080012022-11-17T18:59:28.764376Z"Prinyasart, Thanakorn"https://zbmath.org/authors/?q=ai:prinyasart.thanakorn"Rakbud, Jittisak"https://zbmath.org/authors/?q=ai:rakbud.jittisak"Samphavat, Suchat"https://zbmath.org/authors/?q=ai:samphavat.suchatNeutrabelian algebrashttps://zbmath.org/1496.080022022-11-17T18:59:28.764376Z"Kearnes, Keith A."https://zbmath.org/authors/?q=ai:kearnes.keith-a"Meredith, Connor"https://zbmath.org/authors/?q=ai:meredith.connor"Szendrei, Ágnes"https://zbmath.org/authors/?q=ai:szendrei.agnesAny cogruences \(\alpha\) and \(\beta\) of an algebra in a congruence modular variety satisfy the condition: \(0\leq [\alpha,\beta]\leq\alpha\wedge\beta\). An algebra is called \textit{abelian}, if \([\alpha,\beta]=0\) and is called \textit{neutral} if \([\alpha,\beta]=\alpha\wedge\beta\) [\textit{R. Freese} and \textit{R. McKenzie}, Commutator theory for congruence modular varieties. Cambridge: Cambridge University Press. London: London Mathematical Society (1987; Zbl 0636.08001)].
The authors of the paper introduce the notion of \textit{neutrabelian} algebras in a congruence modular variety. According to their definition, an algebra is \textit{neutrabelian SI} if it is subdirectly irreducible; the centralizer \(\nu\) of the monolith is comparable to all other congruences; for congruences \(\alpha,\beta\leq \nu\), the commutator \([\alpha,\beta]=0\); and \([\alpha,\beta]=\alpha\wedge\beta\), otherwise. An algebra is \textit{neutrabelian} if every its subdirectly irreducible quotient is a neutrabelian SI.
The authors show that a finite algebra in a congruence modular variety is neutrabelian if and only if it has centalizers split at \(0\). As a consequence of this fact they obtain that each finite algebra with cube term and every subalgebra neutrabelian is dualizable. Moreover, a finite algebra in a finitely decidable congruence modular variety is dualizable.
Reviewer: Agata Pilitowska (Warszawa)An algebraic theory of cloneshttps://zbmath.org/1496.080032022-11-17T18:59:28.764376Z"Bucciarelli, Antonio"https://zbmath.org/authors/?q=ai:bucciarelli.antonio"Salibra, Antonino"https://zbmath.org/authors/?q=ai:salibra.antoninoSummary: We introduce the notion of \textit{clone algebra} (\(\mathsf{CA}\)), intended to found a one-sorted, purely algebraic theory of clones. \(\mathsf{CA}\)s are defined by identities and thus form a variety in the sense of universal algebra. The most natural \(\mathsf{CA}\)s, the ones the axioms are intended to characterise, are algebras of functions, called \textit{functional clone algebras} (\(\mathsf{FCA}\)). The universe of a \(\mathsf{FCA}\), called \(\omega\)-\textit{clone}, is a set of infinitary operations on a given set, containing the projections and closed under finitary compositions. The main result of this paper is the general representation theorem, where it is shown that every \(\mathsf{CA}\) is isomorphic to a \(\mathsf{FCA}\) and that the variety \(\mathsf{CA}\) is generated by the class of finite-dimensional \(\mathsf{CA}\)s. This implies that every \(\omega\)-clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to give an answer to a classical question about the lattices of equational theories. The second application is to the study of the category of all varieties of algebras.All unit-regular elements of relational hypersubstitutions for algebraic systemshttps://zbmath.org/1496.080042022-11-17T18:59:28.764376Z"Kunama, P."https://zbmath.org/authors/?q=ai:kunama.pornpimol"Leeratanavalee, S."https://zbmath.org/authors/?q=ai:leeratanavalee.sorasakSummary: Relational hypersubstitutions for algebraic systems are mappings which map operation symbols to terms and map relation symbols to relational terms preserving arities. The set of all relational hypersubstitutions for algebraic systems together with a binary operation defined on this set forms a monoid. In this paper, we determine all unit-regular elements on this monoid of type \(((m), (n))\) for arbitrary natural numbers \(m, n\geq 2\).Solving polynomial fixed point equationshttps://zbmath.org/1496.080052022-11-17T18:59:28.764376Z"Bloom, Stephen L."https://zbmath.org/authors/?q=ai:bloom.stephen-l"Ésik, Zoltán"https://zbmath.org/authors/?q=ai:esik.zoltanFor the entire collection see [Zbl 0825.68120].Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebrashttps://zbmath.org/1496.170152022-11-17T18:59:28.764376Z"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.5|zhang.yi.2|zhang.yi.14|zhang.yi.3|zhang.yi|zhang.yi.1|zhang.yi.10|zhang.yi.6|zhang.yi.8|zhang.yi.4|zhang.yi.12Fix a unitary commutative associative ring \(\mathbf{k}\). A Rota-Baxter algebra is a commutative associative algebra \(R\) over \(\mathbf{k}\) together with a \(\mathbf{k}\)-linear operator \(P: R \longrightarrow R\) satisfying the so-called \textit{Rota-Baxter} identity for \(f, g\) in \(R\):
\[
P(f)P(g) = P(fP(g)) + P(P(f)g).
\]
This is a special case of a more general definition in the paper. We make this simplification in the review because the main results of the paper deal with the special case. As a pioneering example, the ring \(\mathrm{Cont}(\mathbb{R})\) of continuous functions on \(\mathbb{R}\) is a Rota-Baxter algebra over \(\mathbb{R}\), with the operator \(P\) defined by the Riemann integral for \(f \in \mathrm{Cont}(\mathbb{R})\) and \(x \in \mathbb{R}\):
\[
(P(f))(x) := \int_0^x f(t)\, dt.
\]
Let \(\Omega\) be an index set. A matching Rota-Baxter algebra with respect to \(\Omega\) is a commutative associative algebra \(R\) together with a family of linear operators \(P_{\alpha}: R \longrightarrow R\) indexed by \(\alpha \in \Omega\) such that for \(x, y\) in \(R\) and \(\alpha, \beta\) in \(\Omega\) we have:
\[
P_{\alpha}(x) P_{\beta}(y) = P_{\alpha}(xP_{\beta}(y)) + P_{\beta}(P_{\alpha}(x)y).
\]
Each pair \((R, P_{\alpha})\) for a fixed \(\alpha\) forms an ordinary Rota-Baxter algebra. If \((R, P)\) is a Rota-Baxter algebra and \((g_{\alpha})_{\alpha \in \Omega}\) is a family of elements of \(R\), then we can equip \(R\) with a structure of matching Rota-Baxter algebra by setting \(P_{\alpha}(f) := P(g_{\alpha} f)\) for \(\alpha \in \Omega\) and \(f \in R\).
Let \(\mathbf{CAlg}_{\mathbf{k}}\) denote the category of commutative associative algebras over \(\mathbf{k}\) and \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) denote the category of matching Rota-Baxter algebras. By definition we have a forgetful functor \(\mathcal{F}: \mathbf{MRBA}_{\mathbf{k},\Omega} \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\).
One of the main results of this paper is an explicit construction, via \textit{shuffle product}, of a functor \(\mathcal{G}: \mathbf{CAlg}_{\mathbf{k}} \longrightarrow \mathbf{MRBA}_{\mathbf{k},\Omega}\) left adjoint to \(\mathcal{F}\). In more details, for \(A\) a commutative associative algebra, \(\mathcal{G}(A)\) is the tensor product algebra of \(A\) with the shuffle algebra associated to the \(\mathbf{k}\)-module \(\mathbf{k}\Omega \otimes A\), which is naturally a matching Rota-Baxter algebra. The authors also extend this result to the relative setting. Fix an object \(X\) of \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) and let \(\mathcal{C}_X\) denote the category of morphisms \(X \longrightarrow Y\) in \(\mathbf{MRBA}_{\mathbf{k},\Omega}\). Then the forgetful functor \(\mathcal{C}_X \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\) sending a morphism \(X \longrightarrow Y\) to \(\mathcal{F}(Y)\) is shown to admit an explicit left adjoint.
Reviewer: Huafeng Zhang (Villeneuve d'Ascq)Nullstellensatz for relative existentially closed groupshttps://zbmath.org/1496.200742022-11-17T18:59:28.764376Z"Shahryari, Mohammad"https://zbmath.org/authors/?q=ai:shahryari.mohammad|shahryari.mohammad-reza-baloochSummary: We prove that in every variety of \(G\)-groups, every \(G\)-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize \textbf{Theorem G} of [\textit{G. Baumslag} et al., J. Algebra 219, No. 1, 16--79 (1999; Zbl 0938.20020)]. As a result we see that every pair of \(G\)-existentially closed elements in an arbitrary variety of \(G\)-groups generate the same quasi-variety and if both of them are \(q_\omega\)-compact, they are geometrically equivalent.Green's relations on regular elements of semigroup of relational hypersubstitutions for algebraic systems of type \(((m),(n))\)https://zbmath.org/1496.201122022-11-17T18:59:28.764376Z"Leeratanavalee, Sorasak"https://zbmath.org/authors/?q=ai:leeratanavalee.sorasak"Daengsaen, Jukkrit"https://zbmath.org/authors/?q=ai:daengsaen.jukkritSummary: Any relational hypersubstitution for algebraic systems of type
\[
(\tau,\tau')=((m_i)_{i\in I},(n_j)_{j\in J})
\]
is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any \(n_j\)-ary relational symbol to an \(n_j\)-ary relational term preserving arities, where \(I,J\) are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by \textit{J. Koppitz} and \textit{D. Phusanga} [``The monoid of hypersubstitutions for algebraic systems'', J. Announc. Union Sci. Sliven 33, No. 1, 120--127 (2018)]. In this paper, we study the Green's relations on the regular part of this monoid of a particular type \((\tau,\tau')=((m),(n))\), where \(m,n\geq 2\).Lazy groupoidshttps://zbmath.org/1496.201312022-11-17T18:59:28.764376Z"Kaprinai, Balázs"https://zbmath.org/authors/?q=ai:kaprinai.balazs"Machida, Hajime"https://zbmath.org/authors/?q=ai:machida.hajime"Waldhauser, Tamás"https://zbmath.org/authors/?q=ai:waldhauser.tamasA binary algebraic structure, or groupoid, \((G,*)\), is called lazy if all operations obtained by composition are \(x*y\), \(x*x\) or \(x\) (up to renaming of variables). For example, rectangular bands are lazy. An essentially unary operation \(x*y=f(x)\) is lazy if and only if \(f^2(x)=x\) or \(f^2(x)=f(x)\).
The paper characterizes lazy groupoids in two ways. It describes all 15 maximal varieties of lazy groupoids and their subvarieties, providing explicit equational bases. For each maximal variety of lazy groupoids, there is a universal construction of all members of the variety, similar in spirit to the classical description of rectangular bands.
Reviewer: David Stanovsky (Praha)Fuzzy bounded operators with application to Radon transformhttps://zbmath.org/1496.460792022-11-17T18:59:28.764376Z"Bînzar, Tudor"https://zbmath.org/authors/?q=ai:binzar.tudor"Pater, Flavius"https://zbmath.org/authors/?q=ai:pater.flavius-lucian"Nădăban, Sorin"https://zbmath.org/authors/?q=ai:nadaban.sorin-florinSummary: This paper is focused on developing the means to extend the range of application of the inverse Radon transform by enlarging the domain of definition of the Radon operator, namely from a specific Banach space to a more general fuzzy normed linear space. This is done by studying different types of fuzzy bounded linear operators acting between fuzzy normed linear spaces. The motivation for considering this type of spaces comes from the existence of an equivalence between the probabilistic metric spaces and fuzzy metric spaces, in particular fuzzy normed linear spaces. We mention that many notions and results belonging to classical metric spaces could also be found in this general context. Moreover, this setup allows to develop applications as diverse as: image processing, data compression, signal processing, computer graphics, etc. The class of operators that best fits the intended purpose is the class of strongly fuzzy bounded linear operators. The main results about this family of operators use the fact that the space of such operators becomes a normed algebra. An extension of the classical norm of a bounded linear operator between two normed spaces to the norm of strongly fuzzy bounded linear operators acting between fuzzy normed linear spaces is proved. A version of the classical Banach-Steinhaus theorem for strongly fuzzy bounded linear operators is given. A sufficient condition for the limit of a sequence of strongly fuzzy bounded linear operators to be strongly fuzzy bounded is shown. The adjoint operator of a strongly fuzzy bounded linear operator is a classic bounded linear operator. The class of neighborhood fuzzy bounded linear operators are studied as well, being established connections with two other classes of operator, namely the class of fuzzy bounded linear operators and strongly fuzzy bounded linear operators.The power of the combined basic linear programming and affine relaxation for promise constraint satisfaction problemshttps://zbmath.org/1496.682552022-11-17T18:59:28.764376Z"Brakensiek, Joshua"https://zbmath.org/authors/?q=ai:brakensiek.joshua"Guruswami, Venkatesan"https://zbmath.org/authors/?q=ai:guruswami.venkatesan"Wrochna, Marcin"https://zbmath.org/authors/?q=ai:wrochna.marcin"Živný, Stanislav"https://zbmath.org/authors/?q=ai:zivny.stanislav