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Register a new userOn the Rank Functions of Powerful Sets
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Benjamin Jones
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A set $Ssubseteq 2^E$ of subsets of a finite set $E$ is emph{powerful} if, for all $Xsubseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we call the rank function. Powerful sets were introduced by Farr and Wang as a generalisation of binary matroids, as the cocircuit space of a binary matroid gives a powerful set with the corresponding matroid rank function. In this paper we investigate how structural properties of a powerful set can be characterised in terms of its rank function. Powerful sets have four types of degenerate elements, including loops and coloops. We show that certain evaluations of the rank function of a powerful set determine the degenerate elements. We introduce powerful multisets and prove some fundamental results on them. We show that a powerful set corresponds to a binary matroid if and only if its rank function is subcardinal. This paper answers the two conjectures made by Farr and Wang in the affirmative.
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An $ntimes n$ matrix $M$ is called a textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,ell} M_{ell,k} = 0$ for every $k e ell$. Dietzfelbinger, Hromkovi{v{c}}, and Schnitger (1996) showed that $n le (mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = binom{mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(mbox{rk} M)^2$.
We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.
An ntimes n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k e ell we have M_{k,ell} M_{ell,k} = 0. Dietzfelbinger, Hromkoviv{c}, and Schnitger (1996) showed that n le (rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on rk M can be improved. We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.
In order theory, a rank function measures the vertical level of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this paper by considering conjugate orders which partition the space of pairwise comparisons of order elements. For us, these elements are real intervals, and we consider the weak and subset interval orders as (near) conjugates. It is also natural to ask about interval rank functions applied reflexively on whatever poset of intervals we have chosen, and thereby a general iterative strategy for interval ranks. We explore the convergence properties of standard and conjugate interval ranks, and conclude with a discussion of the experimental mathematics needed to support this work.
Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X subseteq F^*$. A submonoid $M$ generated by $k$ elements of $A^*$ is $k$-maximal if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X subseteq A^*$ primitive if it is the basis of a $|X|$-maximal submonoid. This extends the notion of primitive word: indeed, ${w}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X subseteq Y^*$. The set $Y$ is therefore called a primitive root of $X$. As a main result, we prove that if a set has rank $2$, then it has a unique primitive root. This result cannot be extended to sets of rank larger than 2. For a single word $w$, we say that the set ${x,y}$ is a {em binary root} of $w$ if $w$ can be written as a concatenation of copies of $x$ and $y$ and ${x,y}$ is a primitive set. We prove that every primitive word $w$ has at most one binary root ${x,y}$ such that $|x|+|y|<sqrt{|w|}$. That is, the binary root of a word is unique provided the length of the word is sufficiently large with respect to the size of the root. Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function $theta$ is defined on $A^*$. In this setting, the notions of $theta$-power, $theta$-primitive and $theta$-root are defined, and it is shown that any word has a unique $theta$-primitive root. This result can be obtained with our approach by showing that a word $w$ is $theta$-primitive if and only if ${w, theta(w)}$ is a primitive set.
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