In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. —————————————

In this paper, we characterize explicitly the separation properties $T_0$ and $T_1$ at a point p in topological category of quantale-valued preordered spaces investigate how these characterizations are related. Moreover, prove that local hereditary productive.

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

It is shown that every  almost linear bijection $h : Arightarrow B$ of a unital $C^*$-algebra $A$ onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$-algebra $A$ of real rank zero onto a unital$C^*$-algebra...

Our work is a fundamental study of the notion of approximation inQ-categories and in (U,Q)-categories. Our exposition is categorical but kept as close to language of domain theory as possible. Consequently, we introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize J-continuous Q-categories (...

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

We define natural A∞-transformations and construct A∞-category of A∞-functors. The notion of non-strict units in an A∞-category is introduced. The 2-category of (unital) A∞-categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Sta...

A unital l-group (G,u) is an abelian group G equipped with a translation-invariant lattice-order and a distinguished element u, called orderunit, whose positive integer multiples eventually dominate each element of G. We classify finitely generated unital l-groups by sequences W = (W0,W1, . . .) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Al...

Let A be a unital separable amenable C∗-algebra and C be a unital C∗-algebra with certain infinite property. We show that two full monomorphisms h1, h2 : A → C are approximately unitarily equivalent if and only if [h1] = [h2] in KL(A,C). Let B be a non-unital but σ-unital C ∗-algebra for which M(B)/B has the certain infinite property. We prove that two full essential extensions are approximatel...

We prove that the infinite tensor power of a unital separable C∗-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C∗-algebras without characters known to lie outside it. This category moreover contains the ...