On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.     

Subalgebra extensions of partial monounary algebras

For a subalgebra B of a partial monounary algebra A we define the quotient partial monounary algebra A/B. Let B, B, C be partial monounary algebras. In this paper we give a construction of all partial monounary algebras A such that B is a subalgebra of A and C ≅ A/B.     

Enveloping algebras of double Poisson-Ore extensions

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.     

格的Fuzy同余理想

文［１］提出了格的Ｆｕｚｚｙ理想、Ｆｕｚｙ同余关系等概念并对其性质进行了探讨本文在此基础上提出了格的Ｆｕｚｚｙ同余理想的概念并分别讨论了格、Ｆｕｚｚｙ商格及格直积的Ｆｕｚｚｙ同余理想，得到了一些初步结果     

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A×DA in terms of the combinatorics of the quiver and relations is determined.     

Ideal extensions of lattices

Following the well-known Schreier extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford-Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice L and a lattice K having a least element, and construct (all) the lattices V which have an ideal L′ which is isomorphic to L and the Rees quotient V|L′ is isomorphic to K. Conversely, we prove that each lattice which is an extension of L by K can be so constructed. An illustrative example is given at the end. The text was submitted by the author in English.     

Central extensions of some Lie algebras

We consider three Lie algebras: , the Lie algebra of all derivations on the algebra of formal Laurent series; the Lie algebra of all differential operators on ; and the Lie algebra of all differential operators on We prove that each of these Lie algebras has an essentially unique nontrivial central extension.

     

Contractors and connectors of graph algebras

We study generalizations of the “contraction‐deletion” relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman at al [Reflection positivity, rank connectivity, and homomorphisms of graphs, J. Amer. Math. Soc. 20 (2007), 37–51.] Graph algebras are defined by a graph parameter, and they were introduced in order to study and characterize homomorphism functions. We prove that for homomorphism functions, these graph algebras have special elements called “contractors” and “connectors”. This gives a new characterization of homomorphism functions. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 11–30, 2009     

PMS-代数中的正则理想

本文提出伪补 Ｍ Ｓ 代数（简称 Ｐ Ｍ Ｓ 代数）中正则理想、正则同余关系的概念,研究正则理想与核理想、０ 理想之间的关系,讨论正则同余关系的性质,得到若干结果     

分配伪补格的主理想

本文给出了分配伪补格 ( L;∧ ,∨ ,* ,0 ,1 )中的主理想 I=( d]成为同余理想的充分必要条件 .当 L是局部有限时 (即 d∈ S( L) ,Fd={x|x* * =d}有限 ) ,对骨架 S( L)中的每个元素 d,我们找到了以 I=( d]为核心的最小同余关系 ,利用以上结果我们得到一个 Stone代数是布尔代数的一些等价条件 .     

Graded-division algebras and Galois extensions

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras.On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field $\mathbb{F}$ is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of $\mathbb{F}$ and the isomorphism class of a G-Galois extension of $\mathbb{F}$.This connection is used to classify the simple G-Galois extensions of $\mathbb{F}$ in terms of a Galois field extension $\mathbb{L}/\mathbb{F}$ with Galois group isomorphic to a quotient $G/K$ and an element in the quotient ${\mathrm{\text{Z}}}^2(K,{\mathbb{L}}^{\times })/{\mathrm{\text{B}}}^2(K,{\mathbb{F}}^{\times })$ subject to certain conditions. Non-simple G-Galois extensions are induced from simple T-Galois extensions for a subgroup T of G. We also classify finite-dimensional G-graded-division algebras and, as an application, finite G-graded-division rings.     

Dorroh extensions of algebras and coalgebras

We study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.     

软代数的素理想与同余关系

本文用软代数的素理想集刻划了软代数的每一个同余关系，并给出了软代数的同余理想用素理想集刻划的表达式。     

Full duality among graph algebras and flat graph algebras

We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non--dualizable for every infinite cardinal . A new, general method for proving strong duality is presented. Received August 30, 1999; accepted in final form September 22, 1999.     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.