Interval-valued fuzzy k-ideals and k-regularity of semirings

In this paper, we characterize k-regularity of semirings and study some important properties of k-regularity of semirings in terms of interval-valued fuzzy k-ideals of semirings.     

On Bipolar Anti Fuzzy h-ideals in Hemi-rings

In this work, we present notions of bipolar anti fuzzy $h$-ideals and bipolar anti fuzzy interior $h$-ideals in hemi-rings. Investigating some of their properties, we characterize hemi-rings by means of positive anti $\beta$-cut and negative anti $\alpha$-cut. Meanwhile, some results of homomorphisms, anti images and anti pre-images are given to show the rationality of the definitions introduced in the present paper. Also, we define an equivalence relation on bipolar anti fuzzy $h$-ideals. In particular, we investigate translations, extensions and multiplications of bipolar anti fuzzy $h$-ideals. Finally, we present characterizations of h-hemi-regular and h-semi-simple hemi-rings in terms of bipolar anti fuzzy $h$-ideals.     

<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     

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.     

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     

Special classes of <Emphasis Type="Italic">l</Emphasis>-rings

We study a special class of lattice-ordered rings and a special radical. We prove that a special radical of an l-ring is equal to the intersection of the right l-prime l-ideals for each of which the following condition holds: the quotient l-ring by the maximal l-ideal contained in a given right l-ideal belongs to the special class. The prime radical of an l-ring is equal to the intersection of the right l-semiprime l-ideals. We introduce the notion of a completely l-prime l-ideal. We prove that N ₃(R) is equal to the intersection of the completely l-prime, right l-ideals of an l-ring R, where N ₃(R) is the special radical of the l-ring R defined by the class of l-rings without positive divisors of zero.     

Radicals and <Emphasis Type="Italic">l</Emphasis>-modules

In this paper, we introduce the notion of an l-prime l-ideal and that of a right l-semiprime l-ideal. We prove that our definitions coincide with the definitions of M. A. Shatalova in the case of two-sided l-ideals. Our main results are the following ones. The radical of an l-ring can be represented as the intersection of the right l-ideals for each of which the following condition holds: the quotient ring by the maximal l-ideal contained in the given right l-ideal is semisimple. The hypernilpotent radical of an l-ring can be represented as the intersection of the right l-semiprime ideals satisfying the same condition.     

Representation of normal bands as semigroups of <Emphasis Type="Italic">k</Emphasis>-bi-ideals of a semiring

Here we show that every normal band N can be embedded into the normal band \(\mathcal {B(S)}\) of all k-bi-ideals, the left part \(N/ \mathcal {R}\) of N into the left normal band \(\mathcal {R(S)}\) of all right k-ideals, the right part \(N/ \mathcal {L}\) of N into the right normal band \(\mathcal {L(S)}\) of all left k-ideals, and the greatest semilattice homomorphic image \(N/ \mathcal {J}\) of N into the semilattice of all k-ideals of a same k-regular and intra k-regular semiring S.     

Patch-generated Frames and Projectable Hulls

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A. For Bernhard Banaschewski, on the occasion of his 80th birthday.