The following propositions are true: (1) For every natural number n holds 0−′ n = 0. (3)1 Let D be a non empty set, p be a finite sequence of elements of D, and n be a natural number. If 1 ≤ n and n ≤ len p, then p = (p (n−′ 1))a 〈p(n)〉a (p n). Let us observe that every left zeroed add-right-cancelable right distributive left unital commutative associative non empty double loop structure which ...

In this paper, we determine the character space of vector-valued Lipschitz algebra Lipd(X, E), where (X, d) is any metric and E a certain unital semisimple commutative *-Banach algebra.

The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the...

The aim of Coalgebraic Logic is to find formalisms that allow reasoning about T-coalgebras uniformly in the functor T. Moss' seminal idea was to consider the set functor T as providing a modality ∇ T , the semantics of which is given in terms of the relation lifting of T. The latter exists whenever T preserves weak pullbacks. In joint work with Marta Bílková, Alexander Kurz and Jiří Velebil, we...

The concept of a Hopf algebra originated in topology. Classically, Hopf algebras are defined on the basis of unital modules over commutative, unital rings. The intention of the present work is to study Hopf algebra formalism (§1.2) from a universal-algebraic point of view, within the context of entropic varieties. In an entropic variety, the operations of each algebra are homomorphisms, and ten...

Let $R$ be a commutative unital ring, $a\in R$ and $t$ positive integer. $a^{t}$-reduced $R$-modules universally are defined their properties given. Known (resp. new) results about reduced retrieved obtained) by taking $t=1$ rings deduced.

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^2=b,\, \,ab=0 \rangle . \end{aligned}$$ We study recursive construction self-orthogonal codes over I. classify self orthogonal length n size $$2^n$$ (called here quasi self-dual or QSD) up to $$n=6.$$ In particular, we Type IV (QSD...

Let A and B be C*-algebras $$\varphi :A\rightarrow B$$ a $$*$$ -homomorphism. We discuss the properties of kernel (co-)image induced map $$\mathrm {K}_{0}(\varphi ):\mathrm {K}_{0}(A) \rightarrow \mathrm {K}_{0}(B)$$ on level K-theory. In particular, we are interested in case that co-image is torsion free, show it holds when commutative unital, has real rank zero, $$ unital injective. also embe...

where φ : A → B(H) is a unital completely positive linear map from a C-algebra A to linear operators on a Hilbert space H, and x is a self-adjoint element in A with spectrum in I. Subsequently M. D. Choi [3] noted that it is enough to assume that φ is unital and positive. In fact, the restriction of φ to the commutative C-algebra generated by x is automatically completely positive by a theorem ...

Let A be the category of unital C∗-algebras. A counterexample is given to show that the unitary group functor U : A −→ Grp (and hence the composite of the forgetful functor G : Grp −→ Set with U) although being a faithful right adjoint, fails to be algebraic even in the commutative case. In so doing, a canonical presentation of any unital C∗-algebra as a quotient of a free product of copies (i....