Let k be a commutative ring and let C be the operad of differential graded k-modules obtained as the singular k-chains of the linear isometries operad [4, §V.9]. We show that the category of C-algebras is a proper closed model category. We use the amenable description of the coproduct in this category [4, V.3.4] to analyze the coproduct of and develop a homotopy theory for algebras over an arbi...

We prove that AW*-algebras are determined by their projections, their symmetries, and the action of the latter on the former. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-al...

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota–Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graph...

Category theory provides a more abstract and thus more general setting for considering the structure of mathematical objects. 2-dimensional quantum field theories arise in physics as objects that assign vector spaces to 1-manifolds and linear maps to 2-cobordisms. From a categorical perspective, we find that they are the same as commutative Frobenius algebras. Our main goal is to explain this e...

A sovereign monoidal category is an autonomous monoidal category endowed with the choice of an autonomous structure and an isomorphism of monoidal functors between the associated left and right duality functors. In this paper we define and study the algebraic counterpart of sovereign monoidal categories: cosovereign Hopf algebras. In this framework we find a categorical characterization of invo...

In this paper we prove that in prime characteristic there is a functor −p-Leib from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover we define the notion of restricted universal enveloping diassociative algebra Udp(g) of a restricted Leibniz algebra g and we show that Ud...

The purpose of this chapter is to give an introduction to the theory of cluster categories and cluster-tilted algebras, with some background on the theory of cluster algebras, which motivated these topics. We will also discuss some of the interplay between cluster algebras on one side and cluster categories/cluster-tilted algebras on the other, as well as feedback from the latter theory to clus...

Coils as components of Auslander-Reiten quivers of algebras and coil algebras are introduced by Assem and Skowroński. This concept is applied in the present paper to vectorspace categories. The four admissible operations on an Auslander-Reiten component of a vectorspace category, and the notions of v-coils and of vcoil vectorspace categories are introduced. A detailed study on the indecomposabl...

We introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the catego...

in this note, we introduce the concept of a fuzzy filter of a blalgebra,with respect to a t-norm briefly, t-fuzzy filters, and give some relatedresults. in particular, we prove representation theorem in bl-algebras. thenwe generalize the notion of a fuzzy congruence (in a bl-algebra) was definedby lianzhen et al. to a new fuzzy congruence, specially with respect to a tnorm.we prove that there i...