On the continuity of lattice isomorphisms on C(X, I)

We investigate topological conditions on a compact Hausdorff space Y, such that any lattice isomorphism φ : C(X, I) → C(Y, I), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of ℕ, such a lattice isomorphism is a homeomorphism.

A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.

The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism.

On Fuzzy Deductive Systems of Hilbert Algebras

In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.

ON SOME RELATIONS BETWEEN THE LATTICES R-nat, R-conat AND R-tors AND THE RINGS THEY CHARACTERIZE

This article consists of two sections. In the first one, the concepts of spanning and cospanning classes of modules, both hereditarily and cohereditarily, are explained, and some closure properties of the class of modules hereditarily cospanned by a conatural class are established, which amount to its being a hereditary torsion class. This gives a function from R-conat to R-tors and it is proven that its being a lattice isomorphism is part of a characterization of bilaterally perfect rings. The second section begins considering a description of pseudocomplements in certain lattices of module classes. The idea is generalized to define an inclusion-reversing operation on the collection of classes of modules. Restricted to R-nat, it is shown to be a function onto R-tors, and its being an anti-isomorphism is equivalent to R being left semiartinian. Lastly, another characterization of R being left semiartinian is given, in terms solely of R-tors.

REFLEXIVE POLYTOPES IN DIMENSION 2 AND CERTAIN RELATIONS IN SL2(ℤ)

It is well known that there are 16 two-dimensional reflexive polytopes up to lattice isomorphism. One can check directly from the list that the number of lattice points on the boundary of such a polytope plus the number of lattice points on the boundary of the dual polytope is always 12. It turns out that two-dimensional reflexive polytopes correspond to certain relations of two generators A and B of SL 2(ℤ) of length 12. We generalize this correspondence to reflexive configurations with winding number w and relations of length 12w.