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Abstract: We characterize Priestley spaces of algebraic, arithmetic, coherent, and Stone frames. As a corollary, we derive the well-known dual equivalences in pointfree topology involving various categories of algebraic frames. PubDate: 2024-12-30
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Abstract: We give a representation of relation algebra $$1896_{3013}$$, which has symmetric atoms $$1'$$, a, b, c, and d. The sole forbidden diversity cycle is bcd; the atom a is flexible. We give a group representation over $$\mathbb {Z}/1531\mathbb {Z}$$. PubDate: 2024-12-30
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Abstract: The aim of this paper is to investigate further properties of z-elements in multiplicative lattices. We utilize z-closure operators to extend several properties of z-ideals to z-elements and introduce various distinguished subclasses of z-elements, such as z-prime, z-semiprime, z-primary, z-irreducible, and z-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where z-elements are closed under finite products and a representation of z-elements in terms of z-irreducible elements in z-Noetherian multiplicative lattices. PubDate: 2024-12-28
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Abstract: Let M be a module over a commutative ring R, and $$\mathcal {R}(_{R}M)$$ denote the complete lattice of radical submodules of M. It is shown that if M is a multiplication R-module, then $$\mathcal {R}(_{R}M)$$ is a frame. In particular, if M is a finitely generated multiplication R-module, then $$\mathcal {R}(_{R}M)$$ is a coherent frame and if, in addition, M is faithful, then the assignment $$N\mapsto (N:M)_{ z }$$ defines a coherent map from $$\mathcal {R}(_{R}M)$$ to the coherent frame $$\mathcal {Z}(_{R}R)$$ of $$ z $$-ideals of R. As a generalization of $$ z $$-ideals, a proper submodule N of M is called a $$ z $$-submodule of M if for any $$x\in M$$ and $$y\in N$$ such that every maximal submodule of M containing y also contains x, then $$x\in N$$. The set of $$ z $$-submodules of M, denoted $$\mathcal {Z}(_{R}M)$$, forms a complete lattice with respect to the order of inclusion. It is shown that if M is a finitely generated faithful multiplication R-module, then $$\mathcal {Z}(_{R}M)$$ is a coherent frame and the assignment $$N\mapsto N_{ z }$$ (where $$N_{ z }$$ is the intersection of all $$ z $$-submodules of M containing N) is a surjective coherent map from $$\mathcal {R}(_{R}M)$$ to $$\mathcal {Z}(_{R}M)$$. In particular, in this case, $$\mathcal {R}(_{R}M)$$ is a normal frame if and only if $$\mathcal {Z}(_{R}M)$$ is a normal frame. PubDate: 2024-12-13
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Abstract: The near-unanimity-closed minions of Boolean functions, i.e., the clonoids whose target algebra contains a near-unanimity function, are completely described. The key concept towards this result is the minorant-minor partial order and its order ideals. PubDate: 2024-11-20
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Abstract: We describe the ordering of a class of clones by minion homomorphisms, also known as minor preserving maps or height 1 clone homomorphisms. The class consists of all clones on finite sets determined by binary relations whose projections to both coordinates have at most two elements. This class can be alternatively described up to minion homomorphisms as the class of multisorted Boolean clones determined by binary relations. We also introduce and apply the concept of a minion core which provides canonical representatives for equivalence classes of clones, more generally minions, on finite sets. PubDate: 2024-11-06
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Abstract: Clones of operations of arity $$\omega $$ (referred to as $$\omega $$-operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity $$\omega $$. More recently, clone algebras have been introduced to study clones of functions, including $$\omega $$-operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity $$\omega $$, which are $$\omega $$-operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity $$\omega $$ and their corresponding invariant relations. Given a Boolean ideal X on the set $$A^\omega $$, we endow the set of $$\omega $$-operations on A with a topology, which we refer to as X-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically with respect to the X-topology. We characterise the X-closed clones of $$\omega $$-operations in terms of $$\textrm{Pol}^\omega $$-$$\textrm{Inv}^\omega $$ and present a method to relate $$\textrm{Inv}^\omega $$-$$\textrm{Pol}^\omega $$ to the classical (finitary) $$\textrm{Inv}$$-$$\textrm{Pol}$$. PubDate: 2024-11-01
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Abstract: We refine and advance the study of the local structure of idempotent finite algebras started in Bulatov (LICS, 2004). We introduce a graph-like structure on an arbitrary finite idempotent algebra including those admitting type 1. We show that this graph is connected, its edges can be classified into 4 types corresponding to the local behavior (set, semilattice, majority, or affine) of certain term operations. We also show that if the variety generated by the algebra omits type 1, then the structure of the algebra can be ‘improved’ without introducing type 1 by choosing an appropriate reduct of the original algebra. Taylor minimal idempotent algebras introduced recently are a special case of such reducts. Then we refine this structure demonstrating that the edges of the graph of an algebra omitting type 1 can be made ‘thin’, that is, there are term operations that behave very similar to semilattice, majority, or affine operations on 2-element subsets of the algebra. Finally, we prove certain connectivity properties of the refined structures. This research is motivated by the study of the Constraint Satisfaction Problem, although the problem itself does not really show up in this paper. PubDate: 2024-10-30
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Abstract: In this paper, type n lattice-ordered algebras are introduced and a characterization is given for those of type 0 and type 1. Moreover we investigate the question: Let A be a lattice-ordered algebra with unit element $$e >0$$ in which every positive element has an inverse. Under what conditions A is lattice and algebra isomorphic to $${\mathbb {R}}$$ ' We have shown that for certain algebras the question has a positive answer, generalizing thus a result of Scheffold. We also obtained a result similar to Edwards’ Theorem for normed lattice-ordered algebras. PubDate: 2024-10-18
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Abstract: In this paper we continue the study of edge-colored graphs associated with finite idempotent algebras initiated in [Bulatov, “Local structure of idempotent algebras I”, CoRR, abs/2006.09599, 2020.]. We prove stronger connectivity properties of such graphs that will allows us to demonstrate several useful structural features of subdirect products of idempotent algebras such as rectangularity and 2-decomposition. PubDate: 2024-10-15
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Abstract: We characterise the slices of the category of graphs that are algebraically universal in terms of the structure of the slicing graph. In particular, we show that algebraic universality is obtained if, and only if, the slicing graph contains one of four fixed graphs as a subgraph. PubDate: 2024-10-04
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Abstract: We generalize to $$\hbox {d}_{\text {0}}$$-algebras a result of Riečanová about the decomposition of a D-lattice by means of a family of central elements. PubDate: 2024-09-24
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Abstract: In the framework of weak congruence lattices, many classes of groups have been characterized up to now, in completely lattice-theoretic terms. In this note, the center of the group is captured lattice-theoretically and nilpotent groups are characterized by lattice properties. PubDate: 2024-09-23
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Abstract: We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins in the lattice. As a byproduct we also obtain that the number of words for the Garside element is given by a little Schröder number. PubDate: 2024-09-17
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Abstract: In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories—whose models are sets equipped with an action by a monoid M—and all hyperaffine theories—whose models are sets with an action by a Boolean algebra B. We improve on Johnstone’s result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine–unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title. PubDate: 2024-09-13
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Abstract: We provide a simple natural duality for the varieties generated by the negation- and implication-free reduct of a finite MV-chain. We study these varieties through the dual equivalences thus obtained. For example, we fully characterize their algebraically closed, existentially closed and injective members. We also explore the relationship between this natural duality and Priestley duality in terms of distributive skeletons and Priestley powers. PubDate: 2024-09-06
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Abstract: A quasivariety $$\mathfrak N$$ is called relative congruence principal if, for every algebra $$A\in \mathfrak N$$, every compact $$\mathfrak N$$-congruence on A is a principal $$\mathfrak N$$-congruence. We characterize relative congruence principal quasivarieties in terms of one identity and two quasi-identities. We will use the characterization to show that there exists a continuum of relative congruence principal quasivarieties of algebras of a signature $$\sigma $$, provided $$\sigma $$ contains at least one operation of arity greater than 1. Several examples are provided. PubDate: 2024-08-27
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Abstract: The override operation $$\sqcup $$ is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions f and g, $$f\sqcup g$$ is the function with domain $${{\,\textrm{dom}\,}}(f)\cup {{\,\textrm{dom}\,}}(g)$$ that agrees with f on $${{\,\textrm{dom}\,}}(f)$$ and with g on $${{\,\textrm{dom}\,}}(g) \backslash {{\,\textrm{dom}\,}}(f)$$. Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature $$(\sqcup )$$. But adding operations (such as update) to this minimal signature can lead to finite axiomatisations. For the functional signature $$(\sqcup ,\backslash )$$ where $$\backslash $$ is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define $$f\curlyvee g=(f\sqcup g)\cap (g\sqcup f)$$ for all functions f and g; this is the largest domain restriction of the binary relation $$f\cup g$$ that gives a partial function. Now $$f\cap g=f\backslash (f\backslash g)$$ and $$f\sqcup g=f\curlyvee (f\curlyvee g)$$ for all functions f, g, so the signatures $$(\curlyvee )$$ and $$(\sqcup ,\cap )$$ are both intermediate between $$(\sqcup )$$ and $$(\sqcup ,\backslash )$$ in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis. PubDate: 2024-08-12
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Abstract: We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value $$\infty $$. We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable. PubDate: 2024-07-09
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