in this paper, a finite element model is developed for the fully hydro-mechanical analysis of hydraulic fracturing in partially saturated porous media. the model is derived from the framework of generalized biot theory. the fracture propagation is governed by a cohesive fracture model. the flow within the fracture zone is modeled by the lubrication equation. the displacement of solid phase, and...

We present the numbers of all non-isomorphic residuated lattices with up to 12 elements and a link to a database of these lattices. In addition, we explore various characteristics of these lattices such as the width, length, and various properties considered in the literature and provide the corresponding statistics. We also present algorithms for computing finite residuated lattices including ...

This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π1 subsets of 2 ω. In both these lattices, any non-minimum element can be split, i.e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, if P >M Q, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A c...

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...

In this paper we extend the notion of  degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and  introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove som...

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ : we call such lattices τ -invariant. These lattices natura...