A completion of an n-lattice L = 〈L,.1, . . . , .n〉 is defined, by analogy with lattices, as a pair 〈e,C〉, where C is a complete n-lattice and e : L → C is an n-lattice embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-lattice. The completion reduces to the Dedekind-MacNeille completion in the dyadic case, which was first formulat...

The least element 0 of a finite meet semi-distributive lattice is a meet of meet-prime elements. We investigate conditions under which the least element of an algebraic, meet semi-distributive lattice is a (complete) meet of meet-prime elements. For example, this is true if the lattice has only countably many compact elements, or if |L| < 2א0 , or if L is in the variety generated by a finite me...

domains (§3.3) are often complete lattices, i.e., algebras over partially ordered sets. We recall in this section the fundamental notions of lattice theory, which are the basis for the soundness of over-approximations of reachable state sets computed by abstract interpretation methods. For further details and proofs we refer to textbooks on lattice theory [DP90] and static analysis [NNH05]. Fro...

Problem 36 of the third edition of Birkhoo's Lattice theory 2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented. Questions regarding the axiomatics of Boolean algebras led Huntington to conjecture , in 1904, that every uniquely complemente...

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore,...

Young’s lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard Young tableaux can be represented as paths in Young’s lattice that go up by one square at each step, and more general paths in Young’s lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri’s rule for multiplying a Schur function by a c...

The notation and terminology used here are introduced in the following papers: [18], [13], [17], [14], [19], [7], [1], [8], [6], [20], [3], [9], [2], [10], [15], [16], [5], [11], [4], and [12]. Let us observe that there exists a lattice which is finite. Let us mention that every lattice which is finite is also complete. Let L be a lattice and let D be a subset of the carrier of L. The functor D...

We construct a complete lattice Z such that the binary supremum function sup : Z × Z → Z is discontinuous with respect to the product topology on Z × Z of the Scott topologies on each copy of Z. In addition, we show that bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X,Z) of continuous functions from X to Z where X may be any topological space and where o...

In this paper, the concept of intuitionistic fuzzy soft sets is applied to residuated lattices. The notion of intuitionistic fuzzy soft filters of a residuated lattice is introduced and some related properties are investigated. Then the representations of generated intuitionistic fuzzy soft filters are established. We prove that the lattice of all intuitionistic fuzzy soft filters of a residuat...

Singular and nonsingular flat bands in a Sierpinski fractal-like photonic lattice are reported. It is demonstrated that the lowest two bands, being isolated degenerate due to geometrical frustration, thus can be spanned by complete set of compact localized states. These states experimentally proven propagate diffractionless lattice. The results reveal interplay between single lattice, pave way ...