Representations of polyadic-like equality algebras

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-\({\mathfrak{m}}\) polyadic equality algebras, where \({\mathfrak{m}}\) is an arbitrary infinite cardinal and \({\mathfrak{m}}\) < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras).     

Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly $\pi$-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element $a\in R$ is called pseudopolar if there exists $p\in R$ such that $p^2=p\in {\mathrm{comm}}^2(a)\text{,}a+p\in U(R)\mathrm{and}{a}^{k}p\in J(R)$ for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly $\pi$-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.     

Pseudo duality in mathematical programming: Unconstrained problems and problems with equality constraints

In this work non-convex programs are analyzed via Legendre transform. The first part includes definitions and the classification of programs that can be handled by the transformation. It is shown that differentiable functions that are represented as a sum of strictly concave and convex functions belong to this class. Conditions under which a function may have such representation are given. Pseudo duality is defined and the pseudo duality theorem for non linear programs with equality constraints is proved.The techniques described are constructive ones, and they enable tocalculate explicitly a pseudo dual once the primal program is given. Several examples are included. In the convex case these techniques enable the explicit calculation of the dual even in cases where direct calculation was not possible.     

Finitely presented solvable groups and Lie algebras with unsolvable equality problem

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 80–92, September, 1989.     

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

On the definition and the representability of quasi‐polyadic equality algebras

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in .     

On the complexity of axiomatizations of the class of representable quasi‐polyadic equality algebras

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omegaUsing games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omega$ and $l< n,$ $k < n$, k′ < ω are natural numbers, then Σ contains infinitely equations in which ? occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Pseudo PCF

We continue our investigation on pcf with weak forms of the axiom of choice. Characteristically, we assume DC+P(Y) when looking at \(\prod\nolimits_{s \in Y} {{\delta _s}} \) . We get more parallels of pcf theorems.     

On soft equality

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we deal with the algebraic structure of soft sets. The lattice structures of soft sets are constructed. The concept of soft equality is introduced and some related properties are derived. It is proved that soft equality is a congruence relation with respect to some operations and the soft quotient algebra is established.     

Isomorphism is equality

The setting of this work is dependent type theory extended with the univalence axiom. We prove that, for a large class of algebraic structures, isomorphic instances of a structure are equal—in fact, isomorphism is in bijective correspondence with equality. The class of structures includes monoids whose underlying types are “sets”, and also posets where the underlying types are sets and the ordering relations are pointwise “propositional”. For monoids on sets equality coincides with the usual notion of isomorphism from universal algebra, and for posets of the kind mentioned above equality coincides with order isomorphism.     

Pseudo Frobenius numbers

For a prime $p$, we call a positive integer $n$ a Frobenius $p$-number if there exists a finite group with exactly $n$ subgroups of order $p^a$ for some $a\ge 0$. Extending previous results on Sylow’s theorem, we prove in this paper that every Frobenius $p$-number $n\equiv 1(mod{p}^2)$ is a Sylow $p$-number, i. e., the number of Sylow $p$-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order $3^a$ for any $a\ge 0$.     

Pseudo Leja sequences

We study pseudo Leja sequences attached to a compact set in the complex plane. The requirements are weaker than those of ordinary Leja sequences, but these sequences still provide excellent points for interpolation of analytic functions and their computation is much easier. We also apply them to the construction of excellent sets of nodes for multivariate interpolation of analytic functions on product sets.     

伪i-内射半模

主要引进了伪i-内射半模的定义,并根据对偶原则,参照k-投射半模及内射模的结论,得到了伪i-内射半模的一些很好的性质,从而实现了把环中内射模的某屿性质在半环中内射半模方面的部分推广.     

One matrix equality

In this article a generalization is given of the results existing in the paper [RZhMat, 1968, 1B712]. In the latter the matrix equality $$A_n = ( - 1)^{\frac{{n + 1}}{2}} \left[ {\left( {\frac{{n - 3}}{2}!} \right)} \right]^2 A_3 ^{\frac{{n - 1}}{2}} + (n - 1)(n - 2)A_{n - 2} ,$$ is derived, where the elements of matrix A_k are certain linear combinations of the interpolation coefficients of the Lagrange central-difference formula for the second derivative with pattern K, and its validity is asserted for n=5, 7, 9, and 11, which can be established by direct calculation. In the present article it is proved that the matrix equality written above holds for any odd n. Matrices of type A_n are encountered when applying the method of lines to certain boundary-value problems in appropriate systems of ordinary differential equations.