The “k = 1” Case of a Problem of Greene and Kleitman from 1976: Join-Irreducible Elements in the Lattice of Sperner 1-Families

Let k ≥ 1. A Sperner k-family is a maximum-sized subset of a finite poset that contains no chain with k + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set Sk(P) of Sperner k-families of a fifinite poset P and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of Sk(P),” adding, “This has apparently not been done even for the case k = 1.”In this article, the case k = 1 is done.

Representation of slim lattice by poset

In this paper, we present the necessary and sufficient conditions for a poset to be a poset of the union of join and meet irreducible elements of the slim lattice. Slim lattices are special finite lattices that are intensively investigated recently. The problem that we solved in this paper is a generalization of the problem proposed very recently by Cz?dli.

Reduction of fuzzy-crisp concept lattice based on order-class matrix

Knowledge reduction is one of critical problems in data mining and information processing. It can simplify the structure of the lattice during the construction of fuzzy-crisp concept lattice. In terms of fuzzy-crisp concept, we develop an order-class matrix to represent extents and intents of concepts, respectively. In order to improve the computing efficiency, it is necessary to reduce the size of lattices as much as possible. Therefore the judgement theorem of meet-irreducible elements is proposed. To deal with attribute reductions, we develop a discernibility Boolean matrix in formal fuzzy contexts by preserving extents of meet-irreducible elements via order-class matrix. A heuristic attribute-reduction algorithm is proposed. Then we extend the proposed model to consistent formal fuzzy decision contexts. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.

Solving Generalized Equations with Bounded Variables and Multiple Residuated Operators

This paper studies the resolution of sup-inequalities and sup-equations with bounded variables such that the sup-composition is defined by using different residuated operators of a given distributive biresiduated multi-adjoint lattice. Specifically, this study has analytically determined the whole set of solutions of such sup-inequalities and sup-equations. Since the solvability of these equations depends on the character of the independent term, the resolution problem has been split into three parts distinguishing among the bottom element, join-irreducible elements and join-decomposable elements.

Congrunce lattices of rectangular lattices1

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.

Atoms in quasilocal integral domains and Cohen–Kaplansky domains

Let [Formula: see text] be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of [Formula: see text]. Special attention is given to the set of atoms in [Formula: see text] and to the existence of atoms in [Formula: see text]. While our main interest is in local Cohen–Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight non-associate atoms having an atom in [Formula: see text].

Magical Realism as Historical Discourse Reflected on Eka Kurniawan's 'Beauty is A Wound'

This study aims to uncover magical realism as historical discourse portrayed in the novel Beauty is A Wound by Eka Kurniawan. Descriptive qualitative method used in this study. The data were collected by reading, classifying and interpreting. The result is the novel Beauty is A Wound has five elements of magical realism as Faris stated, they are; irreducible elements, phenomenal world, the unsettling doubt, merging realms and disruptions of time, space and identity. In relation with magical realism, New Historicism also applied to unearth Indonesia historical discourse since the time of late Dutch colonization, the invasion of Japan, the Independence Era and the New Order Era. Then, the massacre of everyone who were labeled as communist. And the genocide of all the thugs or preman in order to make safer and better society. In this novel, the history of Indonesia was camouflaged and mixed with magical realism because every event that categorized as magical realism led to the past events which related to the history of Indonesia. Keywords: historical discourse, magical realism, new historicism