In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...

Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector sp...

In [2] we investigated the lattice Λ(Df 2) of all subvarieties of the variety Df2 of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice Λ(CA2) of all subvarieties of the variety CA2 of two-dimensional cylindric algebras. We prove that the cardinality of Λ(CA2) is that of the continuum, give a criterion for a subvariety of CA2 to be locally finite, ...

We propose a method to axiomatize by equations the least pre$xed point of an order preserving function. We discuss its domain of application and show that the Boolean modal -calculus has a complete equational axiomatization. The method relies on the existence of a “closed structure” and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation...

motivated by an arens regularity problem, we introduce the concepts of matrix banach space and matrix banach algebra. the notion of matrix normed space in the sense of ruan is a special case of our matrix normed system. a matrix banach algebra is a matrix banach space with a completely contractive multiplication. we study the structure of matrix banach spaces and matrix banach algebras. then we...

We establish a bialgebra theory for anti-flexible algebras in this paper. introduce the notion of an which is equivalent to Manin triple algebras. The study special case bialgebras leads introduction Yang–Baxter equation algebra analogue classical Lie or associative algebra. It unexpected consequence that both and have same form. A skew-symmetric solution gives bialgebra. Finally notions [Formu...

We prove that the category of $\mathbb{Z}_2 ^n$-manifolds has all finite products. Further, we show a ^n$-manifold (resp., ^n$-morphism) can be reconstructed from its algebra global ^n$-functions morphism between ^n$-function algebras). These results are importance in study ^n$ Lie groups. The investigation is more challenging, since completed tensor product structure sheafs two not sheaf. rely...

locally extended affine lie algebras were introduced by morita and yoshii in [j. algebra 301(1) (2006), 59-81] as a natural generalization of extended affine lie algebras. after that, various generalizations of these lie algebras have been investigated by others. it is known that a locally extended affine lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...

In last ten years many mathematicians have studied properties of BL-algebras endowed with a topology. For example A. Di Nola and L. Leustean cite{dn} studied compact representations of BL-algebras, L. C. Ciungu cite{ciu} investigated some concepts of convergence in the class of perfect BL-algebras, J. Mi Ko and Y. C. Kim cite{jun} studied relationships between closure operators and BL-algebras,...