The notion of prime ideals is introduced in transitive BE-algebras. Prime ideals are characterized with the help of principal ideals. Prime ideal theorem is stated and derived for BE-algebras. The concept of minimal prime ideals is introduced in transitive BE-algebras. A decomposition theorem of proper ideals into minimal prime ideals is derived.

In this paper, we introduce the generalized hybrid Tribonacci numbers. These numbers can be considered as a generalization of the generalized complex Tribonacci, generalized hyperbolic Tribonacci and generalized dual Tribonacci numbers. We also obtain some identities for these numbers.

This note is written to show that the definition of the LA- Γ -hypersemi-group and the definition of the ordered LA- Γ -hypersemigroup in [2] should be corrected and that it is not enough to replace the “Γ” of the ordered LA-Γ-semigroup by “◦ Γ ◦” to pass from an ordered LA- Γ -semigroup to an ordered LA- Γ -hypersemigroup. The definition of the (m,n)- Γ -hyperideal and the strange symbols used throughout the paper have no sense as well.

Since the reduct of every residuated lattice is a semiring, we can ask under what condition a semiring can be converted into a residuated lattice. It turns out that this is possible if the semiring in question is commutative, idempotent, G-simple and equipped with an antitone involution. Then the resulting residuated lattice even satisfies the double negation law. Moreover, if the mentioned semiring is finite then it can be converted into a residuated lattice or join-semilattice also without asking an antitone involution on it. To a residuated lattice L which does not satisfy the double negation law there can be assigned a so-called augmented semiring. This can be used for reconstruction of the so-called core C(L) of L. Conditions under which C(L) constitutes a subuniverse of L are provided.

In the paper, we introduce the notion of a nondistributive ring N as a generalization of the notion of an associative ring with unit, in which the addition needs not be abelian and the distributive law is replaced by n0 = 0n = 0 for every element n of N. For a nondistributive ring N, we introduce the notion of a nondistributive ring of left quotients S−1N with respect to a multiplicatively closed set S ⊆ N, and determine necessary and sufficient conditions for the existence of S−1N.

The main goal of this paper is to investigate the aggregation of diverse families of binary fuzzy relations, fuzzy filters, and fuzzy lattices. Some links between these families and their images via an aggregation are explored.

Let h(x) be a non constant polynomial with rational coefficients. Our aim is to introduce the h(x)-Chebyshev polynomials of the first and second kind Tn and Un. We show that they are in a ℚ-vectorial subspace En(x) of ℚ[x] of dimension n. We establish that the polynomial sequences (hkTn−k)k and (hkUn−k)k, (0 ≤ k ≤ n − 1) are two bases of En(x) for which Tn and Un admit remarkable integer coordinates.

In this paper we give a structural description of completely Archimedean semirings which is an extension of the structure theorem of completely Archimedean semigroups.

In this paper, the concepts of n-fold implicative ideals and n-fold obstinate ideals in BL-algebras are introduced. With respect to this concepts, some related results are given. In particular, it is proved that an ideal is an n-fold implicative ideal if and only if is an n-fold Boolean ideal. Also, it is shown that a BL-algebra is an n-fold integral BL-algebra if and only if trivial ideal {0} is an n-fold obstinate ideal. Moreover, the relation between n-fold obstinate ideals and n-fold (integral) obstinate filters in BL-algebras are studied by using the set of complement elements. Finally, it is proved that ideal I of BL-algebra L is an n-fold obstinate ideal if and only if LT${L \over T}$ is an n-fold obstinate BL-algebra.