206 research outputs found
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts
The arity gap of order-preserving functions and extensions of pseudo-Boolean functions
The aim of this paper is to classify order-preserving functions according to
their arity gap. Noteworthy examples of order-preserving functions are
so-called aggregation functions. We first explicitly classify the Lov\'asz
extensions of pseudo-Boolean functions according to their arity gap. Then we
consider the class of order-preserving functions between partially ordered
sets, and establish a similar explicit classification for this function class.Comment: 11 pages, material reorganize
A binary operation-based representation of a lattice
summary:In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new constructed lattices
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