Ivo G. Rosenberg

An algebraic approach to hyperalgebras

In the past 6 decades the theory of hypergroups and other concrete hyperalgebras has fairly developed but there is still no coherent universal-algebra type theory of hyperalgebras. We represent hyperalgebras on a universe A as special universal algebras on the set P*(A) (of all nonvoid subsets of A), define hyperclones on A and for A finite, study the relationship between the hyperclones on A and the inclusion-isotone clones on P* (A). We introduce new notions of subuniverses, congruences and homomorphisms of hyperalgebras. Finally we raise a few natural problems concerning the lattice of inclusion-isotone clones on P*(A); in particular for the boolean case A={0, 1}.

Multiple-valued hyperstructures

An n-ary hyperoperation on A is a map from A/sup n/ into the set P of nonvoid subsets of A. A hyperclone on A is a set of hyperoperations on A containing all projections and closed with respect to a natural composition. Although special hyperalgebras, like hypergroups, hyperrings etc., have been studied for 6 decades there is no universal-algebra type theory for hyperalgebras. We try to close this gap by embedding hyperoperations on A into the set Q of all /spl sube/-isotone operations on P. The very crucial compatible relations are introduced through this embedding. For A finite we search for a general completeness criterion and the related maximal hyperclones via the maximal subclones of Q. For this we determine the position of Q in the lattice of clones on P and initiate the study of such meet-reducible clones. We find all such clones of the form Q/spl cap/Pol /spl rho/ where /spl rho/ is a proper unary relation on P, toe reduce the case of equivalence relations and show that two types of maximal clones on P produce no maximal subclone of Q.

Sperner properties for groups and relations

An equivalence on the family of subsets of an e-element set E is hereditary if |a| = |b| and | x {⊆ a : x ∼}|=| x {⊆ b : x ∼}| whenever a , b , c , ⊂ E and a ˜ b. Let W i ˜ denote the number of blocks of ˜ consisting of i -element sets. Setting e =⌊1/2 e ⌋ we prove W 0 ~ ≤ ⋯ ≤ W e ~ and W p ~ ≤ W e − p ~ for all p ≼ e ′. The equivalence ˜ is symmetric ( is selfdual ) if W p ~ = W e − p ~ for all p (if a ∼ b ⇔ E \ a ∼ E \ b ). We prove ˜ is symmetric if ˜ is selfdual. The set of blocks of ˜ has a natural order with X ≼ Y if x ⊆ y for some x ∈ X and y ∈ Y . We study the properties of this order, in particular, we prove that for ˜ symmetric the order has the strong Sperner property: for all k the union of the k largest levels is a maximum sized k-family (i.e. a maximum sized union of k antichains). For a permutation group G on E put a ˜ G b if b = g(a) for some g ∈ G . This set-orbit partition is symmetric and therefore the associated order has the strong Sperner property. A direct application proves that the following finite orders have the strong Sperner property: (a) product of chains ( De Bruijn et al., 1949 ) and (b) the initial segments of the product of two chains ( Stanley, 1980 ). Another consequence is that among the unlabelled graphs on n vertices the graphs with ⌊ 1 2 ( n 2 ) Ȱ edges form a maximum sized family allowing no embedding (as a subgraph) between its members. For a binary relation R set a ˜ R b if R ⋂ a 2 (i.e. the restriction of R to the set a ) is isomorphic to R ⋂ b 2 . This equivalence is hereditary. Its equivalence classes are essentially the isomorphism types of restrictions of R and the above order is the usual embedability order of isomorphism types. A consequence of the main result is that for a homogeneous R the order has the strong Sperner property.

The number of orthogonal permutations

A problem on maximal clones in universal algebra leads to the natural concept of orthogonal orders and their characterization. Two (partial) orders on the same set P are orthogonal if they share only trivial endomorphisms, i.e. if the identity self-map of P is the sole non-constant self-map preserving (i.e. compatible with) both orders. We start with a neat and easy characterization of orthogonal pairs of chains (i.e. linear or total orders) and then proceed to the study of the number q(k) of chains on {0, 1, …, k − 1} orthogonal to the natural chain 0 < 1 < ⋯ < k − 1. We obtain a recurrence formula for q(k) and prove that the ratio q(k)k! (of such chains among all chains) goes to e−2 = 0.1353 ⋯ as k → ∞. Results are formulated in terms of permutations.

Partial Sheffer Operations

A partial operation f on a finite set A is called Sheffer if the partial algebra 〈 A, f 〉 is complete, i.e. if all partial operations on A are compositions of f (or definable in terms of f ). In this paper we describe all partial Sheffer operations for |A| = 2 and all binary Sheffer operations for |A| = 3.

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial g, we can obtain a Bernstein polynomial of degree m with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0,1m,2m,...,1, provided that m is sufficiently large. The second result pertains to a subset of Bernstein polynomials: those with coefficients that are all in the unit interval. We show that polynomials in this subset map the open interval (0,1) into the open interval (0,1) and map the points 0 and 1 into the closed interval [0,1]. The motivation for this work is our research on probabilistic computation with digital circuits. Our design methodology, called stochastic logic, is based on Bernstein polynomials with coefficients that correspond to probability values; accordingly, the coefficients must be values in the unit interval. The mathematics presented here provides a necessary and sufficient test for deciding whether polynomial operations can be implemented with stochastic logic.

A projection property

E. Corominas introduced recently this notion for posets:P is projective if every mapf fromP2 toP which is order preserving and idempotent (i.e.f(x, x)=x for allx ε P) is a projection. We consider extensions of this notion to other structures, as well as to maps withn variables. We prove thatn-projectivity forn≥2 is equivalent to 2-projectivity, with a single exception: the structure has the same morphisms as the collection of congruences associated with a vector space over ℤ/2, of dimension at least two. Focusing on relational structures, Arrows Theorem is introduced as an example. We consider particular types of relational structures: posets, graphs and metric spaces, and discuss for these the specific examples of crowns, cycles and circles.

Intersections of isotone clones on a finite set

Let <or= be a fixed order of height at least 2 on a set A (i.e. contains a chain a<b<c). It is shown that all the isotone clones preserving orders on A isomorphic to <or= intersect in the clone K/sub A/ of trivial functions (i.e. all the projections and all the constant operations on A). It is further shown that for A finite with at least eight elements and for any six-element set there exist two orders on A such that every joint endomorphism is trivial (i.e. id/sub A/ or constants). The same is true for intersections of isotone clones. This yields that with the above restrictions there are four maximal isotone clones intersecting in K/sub A/. Separate considerations are given on the intersections of maximal isotone clones for mod A mod =3 and 4.<<ETX>>

MAXIMAL PARTIAL CLONES DETERMINED BY THE AREFLEXIVE RELATIONS

Abstract Completeness or primality for partial algebras on a finite universe A is defined in a way similar to full algebras. A universal completeness criterion reduces to finding the complete list of maximal partial clones. To each such clone C (with one exception, in fact a non-strong maximal clone) there is an h-ary relation on A so that C consists of all partial operations ⨍ admitting ϱ as subalgebra of 〈A,⨍〉h. In this paper we determine all such ϱ that are areflexive (i.e. consist of repetition free h-tuples). These relations are described combinatorially as relations admitting a special strong coloring. For h = 2 we obtain exactly bipartite graphs and directed graphs without two consecutive arcs. For h = 3 and ϱ totally symmetric the problem of deciding whether such coloring exists is known to be NP-complete.

Joint canonical fuzzy numbers

Abstract Starting from Ramik and Nakamuras report (1991) we introduce and fully characterize the joint canonical fuzzy numbers of arbitrary dimension. The crucial part is to find the least value of the maximum of absolute values of several linear combinations. Surprisingly, there is a formula for it based upon determinants formed by the coefficients of the linear combinations. Finally we give a full description of joint canonical fuzzy numbers of dimension 2 provided by 5 infinite families.