Several noteworthy classes of Boolean functions can be characterized by algebraic identities (e.g. the class of positive functions consists of all functions f satisfying the identity f(x) ∨ f(y) ∨ f(x ∨ y) = f(x ∨ y)). We give algebraic identities for several of the most frequently analyzed classes of Boolean functions (including Horn, quadratic, supermodular, and submodular functions) and proceed then to the general question of which classes of Boolean functions can be characterized by algebraic identities. We answer this question for function classes closed under addition of inessential (irrelevant) variables. Nearly all classes of interest have this property. We show that a class with this property has a characterization by algebraic identities if and only if the class is closed under the operation of variable identiÿcation. Moreover, a single identity su ces to characterize a class if and only if the number of minimal forbidden identiÿcation minors is ÿnite. Finally, we consider characterizations by general ÿrst-order sentences, rather than just identities. We show that a class of Boolean functions can be described by an appropriate set of such ÿrst-order sentences if and only if it is closed under permutation of variables.
Abstract. Pippenger's Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite.
Basic Concepts and TerminologyIn [G] Geiger determined, by explicit closure conditions, the closed classes of endofunctions of several variables (operations) and the closed classes of relations (predicates) on a finite set A. These two dual closure systems are related in a Galois connection given by the "preservation" relation between endofunctions and relations. This Galois theory was also developed independently by Bodnarchuk, Kalužnin, Kotov and Romov in [BKKR]. Removing the finiteness restriction on the underlying set, in [Sz] Szabó characterized the closed classes of endofunctions and closed sets of relations on arbritrary sets. These characterizations involve a local closure property as well as closure under a general scheme of combining families of relations into a new relation, properly extending the schemes described by Geiger in the case of finite sets. Different approaches and formulations, as well as variant There are many natural classes of functions that can not be defined by preservation of a single relation (or preservation of each member of a family of relations), e.g. monotone decreasing functions on an ordered set, or Boolean functions whose Zhegalkin polynomial has degree at most m ≥ 0. However such classes can often be described as consisting of those functions that "transform" one relation to another relation. Also, many natural classes are not classes of endofunctions, the sets in which the function variables are interpreted being different from the codomain of function values, e.g. rank functions of matroids. In the case of finite sets a theory of such functions of several variables, defined as functions from a cartesian product A 1 × . . . × A n of finite sets to a finite set B, was developed by Pöschel in [Pö1]: relations as ordinarily understood are replaced by tuples of relations, then the notion of preservation of relations is naturally extended to such multisorted functions and relational tuples, and the closed classes of functions and relational tuples are determined with respect to the arising Galois connection. Still in the case of finite Date: Final version 01-2005. 1991 Mathematics Subject Classification. 08A02.
Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-Boolean polynomial representations generalize those of monotone, supermodular, and submodular functions. In this paper, these classes are characterized by functional inequalities and are shown to be closed both under algebraic closure conditions and a local closure criterion. It is shown that for every m ≥ 1, in addition to the class of all set functions, there are only three other classes satisfying these algebraic and local closure conditions: those having positive, respectively negative, mth-order derivatives, and those having a polynomial representation of degree less than m.
Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Latticetheoretical properties of decompositions are explored, and connections with particular types of intervals are established.
Chitotriosidase but not angiotensin converting enzyme concentrations correlated with sarcoidosis radiological stages, and also with the degree of lung infiltrate seen by CT-scan, suggesting that the former enzyme (detected locally and sistemically) may play a role in the pathogenesis of the disease. Further studies with a greater number of patients are needed to confirm this hypothesis and to determine whether chitotriosidase may be a marker of the severity of sarcoidosis.
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