Algebra Universalis最新文献

In this article, we give algebraic characterizations of U-frames in terms of ring-theoretic properties of the ring (mathcal {R}L) of real-valued continuous functions on a completely regular frame L. We show that a frame is a U-frame if and only if it is an F-frame and its Čech–Stone compactification is zero-dimensional. We will also introduce frames that are finitely a U-frame and we will characterize them in terms of ring-theoretic properties in (mathcal {R}L).

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.

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}/1531mathbb {Z}).

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.

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 (Nmapsto (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 (xin M) and (yin N) such that every maximal submodule of M containing y also contains x, then (xin 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 (Nmapsto 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.

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.

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.

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}).

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.