Powerset structures of extensional fuzzy sets in sets with similarity relations are investigated. It is proved that extensional fuzzy sets have powerset structures which are powerset theories in the category of sets with similarity relations, and some of these powerset theories are defined also by algebraic theories (monads). Between Zadeh's fuzzy powerset theory and the classical powerset theo...

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E. Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of L. A. Zadeh. On the next step, w...

the paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. motivated by an open question of s. e. rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of l. a. zadeh. on the next step, w...

following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occuring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| ≤ 8000 logN/ log logN for sufficiently large N . The proof combines results from extremal graph theory with numbe...

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation f...

in this paper, let $l$ be a completeresiduated lattice, and let {bf set} denote the category of setsand mappings, $lf$-{bf pos} denote the category of $lf$-posets and$lf$-monotone mappings, and $lf$-{bf cslat}$(sqcup)$, $lf$-{bfcslat}$(sqcap)$ denote the category of $lf$-completelattices and $lf$-join-preserving mappings and the category of$lf$-complete lattices and $lf$-meet-preserving mapping...

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several context...