Let C(X) denote the ring of all real-valued continuous functions on a topological space X and $${\mathcal {R}} (L)$$ R(L) be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring $${\mathcal {R}}_c (L)$$ Rc(L) is introduced as a sub-f-ring of $${\mathcal {R}} ( L)$$ R(L) as a pointfree analogue to the subring $$C_c(X)$$ Cc(X) of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring $${\mathcal {R}}_c (L)$$ Rc(L) . In order to do so we introduce the set $$R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \} $$ Rα:={r∈R:coz(α-r)≠⊤} for every $$\alpha \in {\mathcal {R}} (L)$$ α∈R(L) . We prove that $$R_{\alpha } $$ Rα is a countable subset of $${\mathbb {R}}$$ R for every $$\alpha \in {\mathcal {R}}_c (L)$$ α∈Rc(L) . Next, we show that if L is a compact frame, then $$R_{\alpha } $$ Rα is a finite subset of $${\mathbb {R}}$$ R for every $$\alpha \in {\mathcal {R}}_c (L)$$ α∈Rc(L) . Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and $$C_c(X) \cong C_c(Y )$$ Cc(X)≅Cc(Y) in pointfree topology. Finally, we prove that, for some frame L, the ring $${\mathcal {R}}_c (L)$$ Rc(L) may not be isomorphic to $${\mathcal {R}} (M)$$ R(M) , for any given frame M.

The aim of this article is to propose an adequate completion for distributive nearlattices. We give a proof of the existence of such a completion through a representation theorem, which allows us to prove that this completion is a completely distributive algebraic lattice. We show several properties about this completion, and we present a connection with the free distributive lattice extension of a distributive nearlattice. Finally, we consider how can be extended n-ary operations on distributive nearlattices, and we study the basic properties of these extensions.

We show that if $${\mathsf {V}}$$ V is a semigroup pseudovariety containing the finite semilattices and contained in $$\mathsf {DS}$$ DS , then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of $${\mathsf {J}}$$ J -reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that $$\mathsf {DH}\cap \mathsf {ECom}$$ DH∩ECom is local, for any group pseudovariety $${\mathsf {H}}$$ H .

By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of countable Heyting algebras. We show that this characterization no longer holds in the uncountable case. Our results have analogues for co-Heyting algebras and bi-Heyting algebras, and they easily generalize to the setting of p-algebras.

We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras.

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of naturally ordered right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation $$x\sqcap y=(x/y)y$$ x⊓y=(x/y)y is associative and/or commutative. Narhoops with a left unit are proved to have a top element if and only if $$\sqcap $$ ⊓ is commutative, and their congruences are determined by the equivalence class of the left unit. We also show that the four identities defining narhoops are independent.

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and R(L) be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring Rc(L) is introduced as a sub-f-ring of R(L) as a pointfree analogue to the subring Cc(X) of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring Rc(L). In order to do so we introduce the set Ra := {r. R : coz(a- r) = } for every a. R(L). We prove that Ra is a countable subset of R for every a. Rc(L). Next, we show that if L is a compact frame, then Ra is a finite subset of R for every a. R c(L). Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and Cc(X) = Cc(Y) in pointfree topology. Finally, we prove that, for some frame L, the ring Rc(L) may not be isomorphic to R(M), for any given frame M.

Let L be a finite n-element lattice. We prove that if L has at least $$83\cdot 2^{n-8}$$ 83·2n-8 sublattices, then L is planar. For $$n>8$$ n>8 , this result is sharp since there is a non-planar lattice with exactly $$83\cdot 2^{n-8}-1$$ 83·2n-8-1 sublattices.

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and R (L)R(L) be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring R.sub.c (L)Rc(L) is introduced as a sub-f-ring of R ( L)R(L) as a pointfree analogue to the subring C.sub.c(X)Cc(X) of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring R.sub.c (L)Rc(L). In order to do so we introduce the set R.sub.[alpha] := \ r [member of] R : coz ([alpha] - ) [not equal to] = \R[alpha]:={raR:coz([alpha]-r)a a[currency]} for every [alpha] [member of] R (L)[alpha]aR(L). We prove that R.sub.[alpha] R[alpha] is a countable subset of RR for every [alpha] [member of] R.sub.c (L)[alpha]aRc(L). Next, we show that if L is a compact frame, then R.sub.[alpha] R[alpha] is a finite subset of RR for every [alpha] [member of] R.sub.c (L)[alpha]aRc(L). Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and C.sub.c(X) C.sub.c(Y )Cc(X)aCc(Y) in pointfree topology. Finally, we prove that, for some frame L, the ring R.sub.c (L)Rc(L) may not be isomorphic to R (M)R(M), for any given frame M.

A. Tarski proved that the m-generated free algebra of CA.sub.[alpha] CA[alpha], the class of cylindric algebras of dimension [alpha][alpha], contains exactly 2.sup.m2m zero-dimensional atoms, when m 1ma[yen]1 is a finite cardinal and [alpha][alpha] is an arbitrary ordinal. He conjectured that, when [alpha][alpha] is infinite, there are no more atoms other than the zero-dimensional atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski's conjecture is true if CA.sub.[alpha] CA[alpha] is replaced by D.sub.[alpha] D[alpha], G.sub.[alpha] G[alpha], but the m-generated free Crs.sub.[alpha] Crs[alpha] algebra is atomless.

Convex geometry is a closure space $$(G,\phi )$$ (G,ϕ) with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator $$\phi $$ ϕ of convex geometry $$(G,\phi )$$ (G,ϕ) so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions, for a given convex geometry $$(G,\phi )$$ (G,ϕ) , can be checked in polynomial time in two parameters: the size of the base set |G| and the size of the implicational basis of $$(G,\phi )$$ (G,ϕ) .

We prove that the universal theory and the quasi-equational theory of bounded residuated distributive lattice-ordered groupoids are both EXPTIME-complete. Similar results are proven for bounded distributive lattices with a unary or binary operator and for some special classes of bounded residuated distributive lattice-ordered groupoids.

For a finite lattice Lambda, Lambda-ultrametric spaces have, among other reasons, appeared as a means of constructing structures with lattices of equivalence relations embedding Lambda. This makes use of an isomorphism of categories between Lambda-ultrametric spaces and structures equipped with certain families of equivalence relations. We extend this isomorphism to the case of infinite lattices. We also pose questions about representing a given finite lattice as the lattice of empty set -definable equivalence relations of structures with model-theoretic symmetry properties.

For a finite lattice $$\Lambda $$ Λ , $$\Lambda $$ Λ -ultrametric spaces have, among other reasons, appeared as a means of constructing structures with lattices of equivalence relations embedding $$\Lambda $$ Λ . This makes use of an isomorphism of categories between $$\Lambda $$ Λ -ultrametric spaces and structures equipped with certain families of equivalence relations. We extend this isomorphism to the case of infinite lattices. We also pose questions about representing a given finite lattice as the lattice of $$\emptyset $$ ∅ -definable equivalence relations of structures with model-theoretic symmetry properties.

Define a lattice to be order endoprimal if every order preserving operation on the lattice which is preserved by all endomorphisms is a term operation. We prove that every lattice in the variety generated by $$\mathbf{N}_5$$ N5 is order endoprimal.

We introduce a topological counterpart to the Ponka sums of algebraic structures: the Ponka product of topological spaces. This leads to a duality when considering spaces that are dually equivalent to the algebras used in the construction of the Ponka sum.

It is proved that every prevariety of algebras is categorically equivalent to a ‘prevariety of logic’, i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in the language $$\wedge ,\vee ,\circ $$ ∧,∨,∘ holds in the congruence lattices of all members of every variety of logic, and that being a (pre)variety of logic is not a categorical property.

We introduce a topological counterpart to the PAonka sums of algebraic structures: the PAonka product of topological spaces. This leads to a duality when considering spaces that are dually equivalent to the algebras used in the construction of the PAonka sum.

We prove that for a commutative ring with 1 01a 0, if it is consistently L.sup.*La, then it is consistently O.sup.*Oa. An example is provided to show that a consistently O.sup.*Oa-ring may not be O.sup.*Oa.