Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boolean case |A|=2, the lattice E_A contains uncountably many equational classes, but only countably many of them are clones. The aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its "closed" intervals" [C_1,C_2], for clones C_1 and C_2. For |A|=2, we give a complete classification of all such closed intervals in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of E_A.
The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as intersections of the class of all threshold functions with clones of Boolean functions, and provide a complete classification of such intersections in respect to whether they have finite characterizations. Moreover, we provide a characterizing set of…
Let G be a finite Chevalley group and B a Borel subgroup. Then the interval [B,G] in L(G) is Boolean. We prove, using Zsigmondy's theorem, that for any element P in the open interval (B,G), its lattice-complement P^c is not a group-complement.
Let $P$ be a finite poset. Let $L:=J(P)$ denote the lattice of order ideals of $P$. Let $b_i(L)$ denote the number of Boolean intervals of $L$ of rank $i$. We construct a simple graph $G(P)$ from our poset $P$. Denote by $f_i(P)$ the number of the cliques $K_{i+1}$, contained in the graph $G(P)$. Our main results are some linear equations connecting the numbers $f_i(P)$ and $b_i(L)$. We reprove the Dehn--Sommerville equations for simplicial polytopes. In our proof we use free resolutions and the theory of Stanley--Reisner rings.
We prove a dual version of Øystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient $\hatφ$. For any Boolean group-complemented interval, we observe that $\hatφ = φ\neq 0$ by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that $\hatφ$ is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval $[H,G]$, the graded coset poset $\hat{P} = \hat{C}(H,G)$ is Cohen-Macaulay and the nontrivial reduced Betti number…
In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.
We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetr…
It is shown by the author in 2017 that for the union of N orthants in the n-dimensional space there exists an efficient and systematic way to find the exact value, using a suitable partial order relation construction. In this paper our events are hyperrectangles (or n-orthotopes), the Cartesian product of intervals: another important sets (or events) in both theory and practice. We have discovered a new efficient algorithm for the union of such events. With other important Boolean functions we present optimization problem formulations for both hyperreectangles and orthants.
The purpose of this article is to investigate the combinatorial properties of the cross section lattice of a $J$-irreducible monoid associated with a semisimple algebraic group of one of the types $A_n$, $B_n$, or $C_n$. Our main tool is a theorem of Putcha and Renner which identifies the cross section lattice in the Boolean lattice of subsets of the nodes of a Dynkin diagram. We determine the join irreducibles of the cross section lattice. Exploiting this we find characterizations of the relatively complemented intervals. By a result of Putcha, this determines the Möbius function for $Λ$. We …
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究所）
This paper studies partitions in the space of antimonotonic boolean functions on sets of n elements. The antimonotonic functions are the antichains of the partially ordered set of subsets. We analyse and characterise a natural partial ordering on this set. We study the inter- vals according to this ordering. We show how intervals of antimonotonic functions, and a fortiori the whole space of antimonotonic functions can be partitioned as disjoint unions of certain classes of intervals. These in- tervals are uniquely determined by antimonotonic functions on smaller sets. This leads to recursive e…
Georgia Institute of Technology（佐治亚理工学院）；University of California, Berkeley（加州大学伯克利分校）；Microsoft Research AI（微软研究院人工智能部门）
Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of the associated posets. For function classes arising from matroids, polyhedral cell complexes, and more generally interval Cohen-Macaulay posets, we show that the multigraded Betti numbers are pure, a…
Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, based on a new concept of mixed sets defined as function images of symbol type domains, a compositional approach combining eager and lazy evaluations is proposed. Syntax and semantics are explicitly distinguished. Both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed sym…
