In this paper, we employ the concept of a ζ -ideal 𝒥 and introduce a quotient frame M(X,𝒥) , where X is a topological space. We prove that for a metric space X, this frame is metrizable. Moreover, a necessary and sufficient condition for spatiality of frame M(X,𝒥) will be given. Finally, we apply this general case to the metric space ℝ^n and construct frames which are metrizable but not spatial. More precisely, these frames are derived from the metric space ℝ^n which are actually the quotients of frame 𝒪(ℝ^n) , the set of open subsets of ℝ^n . The technique of constructing these frames, which are denoted by M_i(X, S) , i=0,1,2 , intuitively, is a process in which a given subset of points are removed.
In this paper, we study reflections for ordered universal algebras by means of injective hulls. We show that ordered universal algebras with cut-stable morphisms have a reflection into sup-algebras with residual preserving complete mappings. The reflection gives rise to an order-embedding preserving functor.
Iterating the zero-divisor graph and neighborhood-poset mappings Γ and 𝒩_1 yields a chain that alternates between graphs and posets. This paper examines the periodicity of such chains, with considerable attention given to those that are induced by lattices. For example, if G is a finite connected graph with at least two vertices, and if P is a finite bounded poset containing nonzero zero-divisors, then the induced sequences {(Γ𝒩_1)^n(G)}_n=0^∞ and {(𝒩_1Γ )^n(P)}_n=0^∞ eventually become either stable, or periodic with period equal to 2. In the infinite case, however, a lattice is provided that induces a sequence which fails to become periodic. Also, for every positive integer k, an infinite lattice is provided that induces a periodic sequence with period equal to k.
We show that every finite relatively congruence distributive (RCD) algebra in a quasivariety has a global representation with relatively globally indecomposable factors. As an application, we characterize the congruence permutable finite RCD algebras as those omitting certain homomorphic images. The paper concludes with a case study of the variety of lattices generated by the diamond.
We study the transfer of fundamental order and completeness properties between a truncated Riesz space and its Alexandroff unitization. In particular, we distinguish genuine completeness notions, including relatively uniform completeness, Dedekind completeness, lateral completeness, universal completeness, and the (principal) projection property, from order-theoretic properties such as the Archimedean property. We establish characterizations and equivalences for several of these notions and provide counterexamples to highlight the necessity of certain assumptions and the independence of different completeness properties.
We extend strongly hollow and completely strongly hollow ideals from commutative rings to multiplicative lattices. We characterize these elements through residuals and localizations at maximal elements, and study them in semisimple, Gelfand, Prüfer, and weak r-lattices. Applications include criteria for quasi-locality, descriptions in weak Noether lattices, and representations of multiplicative lattices by completely strongly hollow elements.
We prove that a variety 𝒱 is a Taylor variety if and only if the compatible reflexive antisymmetric digraphs in 𝒱 are cycle-free.
We generalize the notion of the ℱ -limit, initially introduced by Garcia–Ferreira and Ruza–Montilla, to any infinite index set I together with a family ℱ_κ(I) of subsets of I that need not be an ultrafilter. This extension permits the definition of a topology on Spec(R) , called the ℱ_κ(I) -limit topology, whose topological properties we study in detail.
In the category V of unital archimedean vector lattices, several relatively recent results have brought closure to the notion of uniform completion: there are exactly four constructs worthy of the name uniform completion. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the manner in which the convergence is regulated. In each case the complete objects form a full monoreflective subcategory of V , denoted respectively ucV , irucV , orucV , and *𝐜𝐕 . In this article we survey these completions, comparing and contrasting them by means of a novel pointfree variant of the classical Yosida adjunction.
We establish a Cayley-type representation theorem for distributive lattices by constructing an embedding into a suitable endomorphism structure.
We provide a classification for the class of unilinear residuated lattices (URLs) into four natural subclasses. We give axiomatizations for each class and for the varieties they generate. We further produce constructions that show that algebras of each class can be obtained from ⊤ -unital URLs (one of the four classes); this reduces the study of all URLs to the ⊤ -unital ones, already studied in previous work. Finally we show how class operators interact with the constructions and provide descriptions of the subvariety lattices.
An algorithm that decides the uniform word problem for lattices is given and shown to have O(n^3) running-time, which also gives an O(n^3) algorithm for the quasi-equational theory of lattices. This result continues a long sequence of algorithms for this problem, starting with Skolem in 1920. The algorithm makes use of Cosmadakis’ algorithm for the uniform word problem and Freese’s algorithm for the word problem for free lattices.
The variety of weak Heyting algebras corresponds to the strict implication fragment of the normal modal logic K , which is also known as the subintuitionistic local consequence of the class of all Kripke models. In this paper we study the variety of commutative weak Heyting algebras, which is defined as the subvariety of weak Heyting algebras whose members satisfy the identity a→ (b→ c) = b→ (a→ c) . We give a description of the implicative-infimum subreducts of the commutative weak Heyting algebras. Moreover, we present an equivalence for the algebraic category of commutative weak Heyting algebras, which is based in a Priestley-style duality.
Inspired by the well-known Stone’s duality and Priestley’s duality for the class of distributive lattices, Celani and Calomino built a Stone-style duality for the class of distributive nearlattices with greatest element and González presented a Priestley-style duality for the variety of DN-algebras. As we know, the category of distributive nearlattices with greatest element and the category of DN-algebras with greatest element are equivalent, then it is natural to develop the Priestley-style duality for the class of distributive nearlattices with greatest element and consider the relationship between the categories of Stone-like spaces and Priestley-like spaces both for distributive nearlattices with greatest element. In this paper, we shall modify the Stone-style duality for the class of distributive nearlattices with greatest element and present a Priestley-style duality for the class of distributive nearlattices with greatest element. Furthermore, it will be shown that the category of Priestley-like spaces with certain binary relations and the category of Stone-like spaces with certain binary relations are isomorphic.
In this paper, we introduce inverse commutative residuated lattices and study some special ones, namely, totally ordered inverse commutative residuated lattices. After obtaining some properties of such inverse commutative residuated lattices, we establish a structure theorem for totally ordered inverse commutative residuated lattices. As an application, we make use of the structure theorem to prove that the variety generated by all totally ordered inverse commutative residuated lattices is arithmetical, congruence extensible and e-regular.
This paper investigates the existence and structure of square roots in hoops, with a particular focus on cancellative, Wajsberg, and basic hoops that are not necessarily bounded. We introduce novel characterizations and decomposition theorems in this context. Specifically, we show that a cancellative hoop N(G) , arising from the negative cone of an Abelian ℓ -group G , admits a square root μ if and only if G is two-divisible. For Wajsberg hoops with square roots, we introduce the notion of strong strict square roots and prove that any such hoop admits a unique decomposition into a direct product of a strong strict hoop and an idempotent hoop (which is, equivalently, a generalized Boolean algebra), by employing the framework of a nested family of hoops. We also investigate square roots on ordinal sums of hoops and define the ordinal sum of a family of square roots. This approach enables us to characterize linearly ordered hoops admitting square roots. Finally, we extend these results to basic hoops with square roots and identify generators for certain special subvarieties within this class.
We show that the congruence lattice of a semilattice satisfies a form of distributivity relative to principal congruences of the form Θ _t ⊙ s, s. Particularly, we establish that semilattice congruences obey the “pairwise distributive law”: (∩ _i ∈ wΩ _i) ∨Θ _t ⊙ s, s = ∩ _k,r ∈ w ( (Ω _k∩Ω _r) ∨Θ _t ⊙ s, s ) for any two elements t and s and any family of congruences {Ω _i: i∈ w }, with w a possibly infinite set.
In this paper, we generalize the concept of unicoherence to the context of frames. Unicoherence, originally introduced by Kuratowski, is a connectedness property that is well studied in classical topology and used to detect holes of a space. We extend the notion of unicoherence to locales and we then investigate its properties. In particular, we prove that many of the known characterizations of unicoherence for topological spaces extend to the setting of locales. Some of these characterizations interestingly involve separation properties for locales.
