Turning Retractions of an Algebra into an Algebra
نویسنده
چکیده
Dragan Mašulović Abstract. One can turn the set of retractions of a lattice 〈L,≤〉 into a poset Rf (L) by letting f ≤ g iff f(x) ≤ g(x) for all x ∈ L. In 1982 H. Crapo raised the following two problems: (1) Is it true that Rf (L) is a lattice for any lattice L? (2) Is it true that Rf (L) is a complete lattice if L is a complete lattice? In 1990 and 1991 B. Li published two papers dealing with the above two questions. He showed that Rf (L) is not necessarily a lattice and that L is a complete lattice if and only if Rf (L) is a complete lattice. Motivated by the idea of extending the structure from the base set to the set of all retractions, we introduce the notion of R-algebra as follows. Let Rf (A) denote the set of all retractions of an algebra A. We say that A is an R-algebra if the set Rf (A) is closed with respect to operations of A applied pointwise. We give some necessary and some sufficient conditions for A to be an R-algebra. We show that the property of being an Ralgebra carries over to retracts of the algebra. In a set of examples we show that almost no classical algebra is an R-algebra. In particular, a lattice L is an R-algebra iff |L| ≤ 2.
منابع مشابه
Decomposition of ideals into pseudo-irreducible ideals in amalgamated algebra along an ideal
Let $f : A rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we give a necessary and sufficient condition for the amalgamated algebra along an ideal $Abowtie^fJ$ to be $J$-Noetherian. Then we give a characterization for pseudo-irreducible ideals of $Abowtie^fJ$, in special cases.
متن کاملOn subalgebras of an evolution algebra of a "chicken" population
We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. For such algebras in terms of its structure constants we calculate right and plenary periods of generator elements. Some results on subalgebras of EACP and ideals on low-dimensional EACP are obtained.
متن کاملCharacterization of n–Jordan homomorphisms on Banach algebras
In this paper we prove that every n-Jordan homomorphis varphi:mathcal {A} longrightarrowmathcal {B} from unital Banach algebras mathcal {A} into varphi -commutative Banach algebra mathcal {B} satisfiying the condition varphi (x^2)=0 Longrightarrow varphi (x)=0, xin mathcal {A}, is an n-homomorphism. In this paper we prove that every n-Jordan homomorphism varphi:mathcal {A} longrightarrowmathcal...
متن کاملDecomposition of H*-Algebra Valued Negative Definite Functions on Topological *-Semigroups
In the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup S with a weight function w into a proper H*-algebra A in terms of w-bounded continuous positive definite A-valued functions on S. A generalization of a well-known result of K. Harzallah is obtained. An earlier conjecture of the author ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005