Some Ideal Lattices in Partial Abelian Monoids and Effect Algebras

Congruences and ideals in partial Abelian monoids (PAM) are studied. It is shown that the so-called R ₁-ideals in cancellative PAMs (CPAM) form a complete Brouwerian sublattice of the lattice of all ideals, and they are standard elements of it. In a special class of CPAMs, effect algebras, properties of ideals and congruences are studied in relation to the generalized Sasaki projections and dimensional equivalence.     

Implicative hyper <Emphasis Type="Italic">K</Emphasis>-algebras

In this note we first define the notions of (weak, strong) implicative hyper K-algebras. Then we show by examples that these notions are different. After that we state and prove some theorems which determine the relationship between these notions and (weak) hyper K-ideals. Also we obtain some relations between these notions and (weak) implicative hyper K-ideals. Finally, we study the implicative hyper K-algebras of order 3, in particular we obtain a relationship between the positive implicative hyper K-algebras and (weak, strong) implicative hyper K-algebras under a simple condition.     

格的TL-模糊理想

本文研究了格的TL-模糊理想. 利用生成TL-模糊理想, 证明了一个模格的全体T_M-模糊理想形成一个完备的模格. 此外, 利用L-模糊集的投影和截影, 获得了将直积格的TL-模糊理想表示成分量格的TL-模糊理想的L-直积的一个充分必要条件. 所得结果进一步推广和发展了格的模糊理想的理论.     

Relative polars in ordered sets

In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of R-polars are studied. Connections between R-polars and prime ideals, especially in distributive sets, are found.     

q-Ideals and a-Ideals in BCI-Algebras

In this note we define the notions of q-ideals and a-ideals in BCI-algebras. We give several characterizations and the extensive theorems about q-ideals and a-ideals. We show that a non-empty subset of a BCI-algebra is a-ideal if and only if it is both q-ideal and p-ideal. Finally, we give four characterizations of associative BCI-algebras by a-ideals and eight characterizations of quasi-associative BCI-algebras by q-ideals.Supported by Fujian Education Committee Foundation, China.AMS Subject Classification (2000), 03G25, 06F35     

Ultra <Emphasis Type="Italic">LI</Emphasis>-ideals in lattice implication algebras and <Emphasis Type="Italic">MTL</Emphasis>-algebras

A mistake concerning the ultra LI-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an LI-ideal to be an ultra LI-ideal are given. Moreover, the notion of an LI-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) LI-ideal and an ILI-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper LI-ideal and Boolean LI-ideal, (2) prime proper LI-ideal and ILI-ideal, (3) proper obstinate LI-ideal, (4) ultra LI-ideal. This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605389) and K. C. Wong Magna Fund in Ningbo University.     

<Emphasis Type="Italic">p</Emphasis>-Ideals in <Emphasis Type="Italic">p</Emphasis>-Regular Semirings

In this paper, we introduce p-ideals in semirings. A new form of regularity, which is compatible with p-ideals, is defined. Our aim is to explore the possibilities of establishing an ideal theory in semirings, going alongside the existing literature of semiring theory.AMS Subject Classification (2000): Primary 16Y60     

On L-fuzzy ideals in semirings II

We study some properties of L-fuzzy left (right) ideals of a semiring R related to level left (right) ideals     

Pasting topological spaces at one point

Let {X _α } _α∈Λ be a family of topological spaces and x _α ∈ X _α , for every α ∈ Λ. Suppose X is the quotient space of the disjoint union of X _α ’s by identifying x _α ’s as one point σ. We try to characterize ideals of C(X) according to the same ideals of C(X _α )’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2) Is there any set of prime ideals of cardinality m in a ring R such that the intersection of these prime ideals can not be obtained as an intersection of fewer than m prime ideals in R? Finally, we answer an open question in [11].     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).