the aim of this paper is to extend results established by h. onoand t. kowalski regarding directly indecomposable commutative residuatedlattices to the non-commutative case. the main theorem states that a residuatedlattice a is directly indecomposable if and only if its boolean center b(a)is {0, 1}. we also prove that any linearly ordered residuated lattice and anylocal residuated lattice are d...

The aim of this paper is to extend results established by H. Onoand T. Kowalski regarding directly indecomposable commutative residuatedlattices to the non-commutative case. The main theorem states that a residuatedlattice A is directly indecomposable if and only if its Boolean center B(A)is {0, 1}. We also prove that any linearly ordered residuated lattice and anylocal residuated lattice are d...

$top$-filters can be used to define $top$-convergence spaces in the lattice-valued context. Connections between $top$-convergence spaces and lattice-valued convergence spaces are given. Regularity of a $top$-convergence space has recently been defined and studied by Fang and Yue. An equivalent characterization is given in the present work in terms of convergence of closures of $top$-filters.  M...

This paper explores derivative operations of the Boolean differential calculus for lattices of Boolean functions. Such operations are needed to design circuits with short delay and low power consumption [3] as well as to calculate minimal complete sets of fitting test patterns [4]. It will be shown that each derivative operation of a lattice of Boolean functions creates again a lattice of Boole...

In this note, we study the lattice structure on the class of all weak hyper K-ideals of a hyper K-algebra. We first introduce the notion of (left,right) scalar in a hyper K-algebra which help us to characterize the weak hyper K-ideals generated by a subset. In the sequel, using the notion of a closure operator, we study the lattice of all weak hyper K-ideals of ahyper K-algebra, and we prove a ...

In this paper, we investigate more relations between the symmetric residuated lattices $L$ with their corresponding intuitionistic fuzzy residuated lattice $tilde{L}$. It is shown that some algebraic structures of $L$ such as Heyting algebra, Glivenko residuated lattice and strict residuated lattice are preserved for $tilde{L}$. Examples are given for those structures that do not remain the sam...

This paper explores lattices of Boolean functions which are built by derivative operations of the Boolean Differential Calculus [2], [8], [9]. Such operations are needed to design circuits with short delay and low power consumption [11] as well as to calculate minimal complete sets of fitting test patterns [12]. It will be shown that each derivative operation of a lattice of Boolean functions c...

In a canonical way we establish an AZ{identity (see [9]) and its consequences, the LYM{inequality and the Sperner{property, for the Boolean interval lattice. Further, the Bollobas{inequality for the Boolean interval lattice turns out to be just the LYM{ inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice. Perhaps more surprising is that by our approach t...

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A ⊂ B,C ⊂ D. A diamondfree family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2−o(1)) (...

In this paper we introduce the notion of Boolean filters in a pseudo-complemented distributive lattice and characterize the class of all Boolean filters. Further a set of equivalent conditions are derived for a proper filter to become a prime Boolean filter. Also a set of equivalent conditions is derived for a pseudo-complemented distributive lattice to become a Boolean algebra. Finally, a Bool...