Join-Irreducible Boolean Functions

Abstract

Abstract. This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting posetΩ. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members ofΩ are the −2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members ofΩ.

The Arity Gap of Polynomial Functions over Bounded Distributive Lattices

“…The simple minor relation constitutes a quasi-order ≤ on the set of all B-valued functions of several variables on A which is given by the following rule: f ≤ g if and only if f is obtained from g by simple variable substitution. If f ≤ g and g ≤ f , we say that f and g are equivalent, denoted f ≡ g. If f ≤ g but g ≤ f , we denote f < g. It can be easily observed that if f ≤ g then ess f ≤ ess g, with equality if and only if f ≡ g. For background, extensions and variants of the simple minor relation, see, e.g., [2,5,8,12,13,17,18,21,26,30].…”

“…To show that it is also necessary, assume that for all J ⊆ [n] \ {j}, we have that f (e J ) = f (e J∪{j} ). Consider the DNF representation of f as given by equation (2). Then for every J ⊆ [n] \ {j}, the term f (e J ∪ {j}) ∧ i∈J∪{j} x i is absorbed by the term f (e J ) ∧ i∈J x i .…”

On Functions with a Unique Identification Minor

“…We will show that f is totally symmetric. To this end, it is sufficient to show that Inv f contains all adjacent transpositions (m m + 1), 1 ≤ m ≤ n − 1, i.e., for every m ∈ [n − 1], (2) f (a 1 , . .…”

Content and Singletons Bring Unique Identification Minors

“…This problem was previously posed, although using a somewhat different formalism, in the 2010 paper by Bouaziz, Couceiro, and Pouzet [3, Problem 2 (ii)]. It is well known that the 2-set-transitive functions have a unique identification minor (for a proof of this fact, see [13,Proposition 4.3]; this fact is also implicit in the work of Bouaziz, Couceiro, and Pouzet [3]). It was recently shown by the current author that the functions determined by the order of first occurrence also have this property [15,Corollary 5].…”