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Abstract: A frame is a complete lattice in which the meet distributes over arbitrary joins. Let \(\tau \) be a subframe of a frame L such that every element of \(\tau \) has a complement in L, then \((L, \tau )\) , briefly \(L_{ \tau }\) , is said to be a topoframe. Let \({\mathcal {R}}L_\tau \) be the ring of real-continuous functions on a topoframe \(L_{ \tau }\) . We define P-topoframes and show that \(L_{\tau }\) is a P-topoframe if and only if \({\mathcal {R}}L_{\tau }\) is a regular ring if and only if it is a \(\aleph _0\) -self-injective ring. We define extremally disconnected topoframes and show that \(L_{\tau }\) is an extremally disconnected topoframe if and only if \(\tau \) is an extremally disconnected frame. For a completely regular topoframe \(L_\tau \) , it is shown that \(L_\tau \) is an extremally disconnected topoframe if and only if \({\mathcal {R}}L_\tau \) is a Baer ring if and only if it is a CS-ring. Finally, we prove that a completely regular topoframe \(L_\tau \) is an extremally disconnected P-topoframe if and only if \({\mathcal {R}}L_\tau \) is a self-injective ring. PubDate: 2021-09-30

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Abstract: A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\) , \(d\wedge a\le b\) if and only if \(d\le c\) . This c is denoted by \(a\rightarrow b\) . This pair can be regarded as an algebra \(\left<A,\wedge ,\vee ,\rightarrow ,0,1\right>\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\) . The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices. PubDate: 2021-09-28

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Abstract: An algebra \(\mathbf{A}\) is called a perfect extension of its subalgebra \(\mathbf{B}\) if every congruence of \(\mathbf{B}\) has a unique extension to \(\mathbf{A}\) . This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that \(\mathbf{A}\) is a congruence-preserving extension of \(\mathbf{B}\) . Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra \(\mathbf{M}\) is perfect extension of its Boolean subalgebra \(B(\mathbf{M})\) , the so-called skeleton of \(\mathbf{M}\) . In this note a full solution to this interesting problem is given. The theory of natural dualities in the sense of Davey and Werner [1983] and Clark and Davey [1998], as well as Boolean product representations, are used as the main tools to obtain the solution. PubDate: 2021-09-18

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Abstract: In this paper we investigate measures over bounded lattices, extending and giving a unifying treatment to previous works. In particular, we prove that the measures of an arbitrary bounded lattice can be represented as measures over a suitably chosen Boolean lattice. Using techniques from algebraic geometry, we also prove that given a bounded lattice X there exists a scheme \(\mathcal {X}\) such that a measure over X is the same as a (scheme-theoretic) measure over \(\mathcal {X}\) . We also define the measurability of a lattice, and describe measures over finite lattices. PubDate: 2021-09-16

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Abstract: We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite. PubDate: 2021-09-16

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Abstract: We present a functorial construction which, starting from a congruence \(\alpha \) of finite index in an algebra \(\mathbf {A}\) , yields a new algebra \(\mathbf {C}\) with the following properties: the congruence lattice of \(\mathbf {C}\) is isomorphic to the interval of congruences between 0 and \(\alpha \) on \(\mathbf {A}\) , this isomorphism preserves higher commutators and TCT types, and \(\mathbf {C}\) inherits all idempotent Maltsev conditions from \(\mathbf {A}\) . As applications of this construction, we first show that supernilpotence is decidable for congruences of finite algebras in varieties that omit type \(\mathbf {1}\) . Secondly, we prove that the subpower membership problem for finite algebras with a cube term can be effectively reduced to membership questions in subdirect products of subdirectly irreducible algebras with central monoliths. As a consequence, we obtain a polynomial time algorithm for the subpower membership problem for finite algebras with a cube term in which the monolith of every subdirectly irreducible section has a supernilpotent centralizer. PubDate: 2021-09-07

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Abstract: In the theory of semigroups there exists a construction of some semilattices from a special family of semigroups. It is the so-called ‘strong semilattice of semigroups’. Modifying this idea we can present a new construction method for arbitrary pseudocomplemented semilattice (= PCS) L using ‘full triples’. This construction centers around the classical Glivenko-Frink congruence \(\Gamma (L)\) . This fact plays an important role. Namely, PCS L is a disjoint union of all congruence classes of \(\Gamma (L)\) (or of GF-blocks for short). In order to get a ‘full triple’ of L, the so-called ‘associate’ full triple of L, we need the Boolean algebra of closed elements B(L), the whole family of GF-blocks \(\{\,\Gamma _a\mid a\in B(L)\,\}\) and a suitable semilattice homomorphism \(\varphi _{a,b}:\Gamma _a\rightarrow \Gamma _b\) for any \(a\ge b\) in B(L). There is also a definition of an ‘abstract’ full triple, which we use by a construction of a PCS. The notion of a full triple is an extension of the ‘classical’ triple, which do work only with just one GF-block D(L) satisfying \(1\in D(L)\) . It is known that there exist PCS’s which cannot be constructed by using a classical triple method. In addition, we explore in some detail the homomorphisms and the subalgebras of PCS’s. PubDate: 2021-09-06

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Abstract: We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field \({\mathbb {F}}\) by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal. PubDate: 2021-08-31

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Abstract: The collection of hereditary classes of modules over an arbitrary ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are precisely the natural classes of R-modules. In this paper we extend some results about hereditary classes in R-Mod to the category \(\mathcal {L}_\mathcal {M}\) of linear modular lattices, which has as objects all modular complete lattices and as morphisms all linear morphisms. We also define natural classes in the full subcategory \(\mathcal {L}_{{\mathcal {M}}_{c}}\) of upper semicontinuous modular complete lattices and show that the collection of these classes is the skeleton of the big lattice of hereditary classes in \(\mathcal {L}_{{\mathcal {M}}_{c}}\) and is a boolean big lattice. PubDate: 2021-08-24

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Abstract: A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without top. Results include the following: (a) Every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact \(T_1\) -topology on \(\mathop {\mathrm {max}}L\) , the set of maximal ideals of L, and under weak hypotheses this representation is functorial. (b) Every Wallman base for a topological space is conjunctive; we give an example of a conjunctive annular base that is not Wallman. (c) The free distributive lattice over a conjunctive join-semilattice L is a subsemilattice of the power set of \(\mathop {\mathrm {max}}L\) . (d) For an arbitrary join-semilattice L: if every u-maximal ideal is prime (i.e., the complement is a filter) for every \(u\in L\) , then L satisfies Katriňák’s distributivity axiom. (This appears to be new, though the converse is well known.) If L is conjunctive, all the 1-maximal ideals of L are prime if and only if L satisfies a weak distributivity axiom due to Varlet. We include a number of applications. PubDate: 2021-08-24

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Abstract: The Galois correspondence \({{\mathbf {G}}}\) between sets of logical formulas of the first order language \(\mathbf{L}\) over a signature L and type definable sets over an \(\mathbf{L}\) -structure A on the set \({{\mathbf {A}}}\) is extensively studied in the literature. The Stone topology is successfully applied in model theory. We investigate basic properties of the pointwise convergence topologies in languages \(\mathbf{L_n}(A)\) , that consist of first order formulas with n free variables, and in affine spaces \({{\mathbf {A}}}^n\) over A, and compare these topologies with Stone topology and Zariski topology. In particular, we show, that Zariski topology is strongly weaker than the pointwise convergence topology and the pointwise convergence topology coincides with the Stone topology on a subset U of the first order language \(\mathbf{L}(A)\) if and only if U is finite modulo logic equivalence of formulas. PubDate: 2021-08-05

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Abstract: In this paper we use the theory of central elements in order to provide a characterization for coextensive varieties. In particular, if a variety is of finite type, congruence-permutable and its class of directly indecomposable members is universal, then the variety is coextensive if and only if it is a variety of shells. PubDate: 2021-08-05

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Abstract: It is proven in this note that if non-empty posets A, B, C, and D satisfy \(A^C\cong B^D\) where C, D, and \(A^C\) are finite and connected, then there exist posets E, X, Y, and Z such that \(A\cong E^X\) , \(B\cong E^Y\) , \(C\cong Y\times Z\) , and \(D\cong X\times Z\) . This solves a problem posed by Jónsson and McKenzie in 1982. PubDate: 2021-07-19

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Abstract: We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\) , and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\) -algebras X and Z, along with an intermediate finite \(\mathcal L\) -algebra Y. The decision problem asks whether there are homomorphisms \(g:X\rightarrow Y\) and \(h:Y\rightarrow Z\) such that \(f=hg\) . We show that this problem is NP-complete for most languages. PubDate: 2021-07-12

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Abstract: An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists. PubDate: 2021-06-25

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Abstract: We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation \(x \mapsto -x\) . The primitive operations are \(+\) , \(\vee \) , \(\wedge \) , 0, 1, \(-1\) . A prime example of these structures is \(\mathbb {R}\) , with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation \(x \mapsto \lnot x\) . The primitive operations are \(\oplus \) , \(\odot \) , \(\vee \) , \(\wedge \) , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra \([0, 1]\subseteq \mathbb {R}\) . We obtain the original Mundici’s equivalence as a corollary of our main result. PubDate: 2021-06-25

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Abstract: Let \(\mathbb {S}\) be the commutative and idempotent semiring with additive identity \(\mathbf {0}\) and multiplicative identity \(\mathbf {1}\) . The tropical semiring \(\mathbb {T}\) and the Boolean semiring \(\mathbb {B}\) are common important examples of such semirings. Let \(UT_{n}(\mathbb {S})\) be the semigroup of all \(n\times n\) upper triangular matrices over \(\mathbb {S}\) , both \(UT^{\pm }_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {S})\) be subsemigroups of \(UT_n(\mathbb {S})\) with \(\mathbf {0}\) and/or \(\mathbf {1}\) on the main diagonal, and \(\mathbf {1}\) on the main diagonal respectively. It is known that \(UT_{2}(\mathbb {T})\) is non-finitely based and \(UT^{\pm }_{2}(\mathbb {S})\) is finitely based. Combining these results, the finite basis problems for \(UT_{n}(\mathbb {T})\) and \(UT^{\pm }_{n}(\mathbb {S})\) with \(n=2, 3\) both as semigroups and involution semigroups under the skew transposition are solved. It is well known that the semigroups \(UT^{+}_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {B})\) are equationally equivalent. In this paper, we show that the involution semigroups \(UT^{+}_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {B})\) under the skew transposition are not equationally equivalent. Nevertheless, the finite basis problems for involution semigroups \(UT_n^{+}(\mathbb {S})\) and \(UT_n^{+}(\mathbb {B})\) share the same solution, that is, the involution semigroup \(UT_n^{+}(\mathbb {S})\) is finitely based if and only if \(n=2\) . PubDate: 2021-06-19

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Abstract: We are interested in abstract conditions that characterize homomorphic images of affine quandles. Our main result is a two-fold characterization of this class: one by a property of the displacement group, the other one by a property of the corresponding affine mesh. As a consequence, we obtain efficient algorithms for recognizing homomorphic images of affine quandles, including an efficient explicit construction of the covering affine quandle. PubDate: 2021-06-14

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Abstract: The 6-element Brandt monoid \(B_2^1\) admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of \(B_2^1\) as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of \(B_2^1\) are not finitely based. PubDate: 2021-06-05

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Abstract: Let \(\sum (X)\) be the collection of subrings of C(X) containing \(C^{*}(X)\) , where X is a Tychonoff space. For any \(A(X)\in \sum (X)\) there is associated a subset \(\upsilon _{A}(X)\) of \(\beta X\) which is an A-analogue of the Hewitt real compactification \(\upsilon X\) of X. For any \(A(X)\in \sum (X)\) , let [A(X)] be the class of all \(B(X)\in \sum (X)\) such that \(\upsilon _{A}(X)=\upsilon _{B}(X)\) . We show that for first countable non compact real compact space X, [A(X)] contains at least \(2^{c}\) many different subalgebras no two of which are isomorphic in Theorem 3.8. PubDate: 2021-05-24