<Emphasis Type="Italic">P</Emphasis>-orderings of finite subsets of Dedekind domains

If R is a Dedekind domain, P a prime ideal of R and S⊆R a finite subset then a P-ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101–127, 1997), is an ordering {a _i } _i=1 ^m of the elements of S with the property that, for each 1<i≤m, the choice of a _i minimizes the P-adic valuation of ∏ _j<i (s−a _j ) over elements s∈S. If S, S ^′ are two finite subsets of R of the same cardinality then a bijection φ:S→S ^′ is a P-ordering equivalence if it preserves P-orderings. In this paper we give upper and lower bounds for the number of distinct P-orderings a finite set can have in terms of its cardinality and give an upper bound on the number of P-ordering equivalence classes of a given cardinality.     

含幺逆半群的半格的同余对

本文利用一族含幺逆半群的同余对刻画了其半格的同余对,并给出了含幺逆半群半格的正规同余对族的格与其标准同余对的格之间的同构关系.     

Partial kernel normal systems in regular semigroups

\noindent Let P be an arbitrary full subset of the set E(S) of the idempotents in a regular semigroup S . We shall prove that each congruence on S is completely determined by its partial kernel normal system linking to P and give the kernel normal systems in regular semigroups an abstract characterization. January 19, 1998     

Sectionally Semicomplemented Distributive Nearlattices

A nearlattice S is a meet semilattice together with the property that any two elements possessing a common upper bound have a supremum. Here the authors have introduced the notion of sectionally semicomplemented distributive nearlattices and given several characterizations of them. The skeleton SCon(S) of Con(S), the congruence lattice, consists of all those nearlattice congruences which are the pseudocomplements of members of Con(S). The relationship between skeletal congruences and kernel of skeletal congruences leads to numerous characterizations of sectionally semicomplemented distributive nearlattices and semiboolean algebras. For example we prove, for a distributive nearlattice S with 0, the following conditions are equivalent:

Congruence triples on regular semigroups and their generalizations

In this paper, we introduce the concept of VT-congruence triples on a regular semigroup S and show how such triples can be constructed by using the equivalences on S/ℒ, S/R and the special congruences on S. Also, such congruence triples are characterized so that an associated congruence can be uniquely determined by a given congruence triple. Moreover, we also consider the VH-congruence pairs on an orthocryptogroup.     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

Two-sided networks for completely simple semigroups

Let S be a completely simple semigroup represented as a Rees matrix semigroup M(I,G,P) with normalized sandwich matrix P. On the congruence lattice C(S) of S we consider the relations T _i, K and T _r which identify congruences with the same left trace, kernel and right trace, respectively. These are equivalences whose classes are intervals. The upper and lower ends of these intervals induce the following operators on C(S) T_l, K, T_r, t_l, k and t_r .We construct here the semigroup generated by these operators as a homomorphic image of a semigroup given by generators and relations and demonstrate the minimality of the latter.     

Paired-Domination in <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced P ₅, which seems to be the critical case.     

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

Polynomial algorithms for kernels in comparability, permutation and P 4-free graphs

In this paper, we give three polynomial algorithms which detect a kernel in comparability graphs relatively to an M-orientation, in permutation graphs and in P₄-free graphs with a normal orientation. MSC classification: 05C69, 05C85 Correspondence to: Saoula Youcef     

Congruences and ideals in ternary rings

A ternary ring is an algebraic structure R=(R,t0.1) of type (3, 0, 0) satisfying the identities t(0, x, y) = y = t(x, 0, y) and t(1, x, 0) = x = (x, l, 0) where, moreover, for any a, b, c R there exists a unique d R with t(a, b, d) = c. A congruence on R is called normal if R with t is a ternary ring again. We describe basic properties of the lattice of all normal congruences on R and establish connections between ideals (introduced earlier by the third author) and congruence kernels.     

Banaschewski’s theorem for generalized <Emphasis Type="Italic">MV</Emphasis>-algebras

A generalized MV-algebra A is called representable if it is a subdirect product of linearly ordered generalized MV-algebras. Let S be the system of all congruence relations ϱ on A such that the quotient algebra A/ϱ is representable. In the present paper we prove that the system S has a least element. This work was supported by Science and Technology Assistance Agency under Contract No AVPT-51-032002. The work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information (grant I/2/2005).     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

Some properties of complete congruence lattices

For a complete lattice C, we consider the problem of establishing when the complete lattice of complete congruence relations on C is a complete sublattice of the complete lattices of join- or meet-complete congruence relations on C. We first argue that this problem is not trivial, and then we show that it admits an affirmative answer whenever C is continuous for the join case and, dually, co-continuous for the meet case. As a consequence, we prove that if C is continuous then each principal filter generated by a continuous complete congruence on C is pseudocomplemented. Received January 6, 1998; accepted in final form July 2, 1998.