Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $(x_i,x_j)$ (resp. the $e$-th power of the least common multiple $[x_i,x_j]$) as the $(i,j)$-entry of the matrix by $((x_i, x_j)^e)$ (resp. $([x_i, x_j]^e))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i, x_j]\in S$) for all $1\le i,j \le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer $e\ge 1$, the $n\times n$ power GCD matrix $((x_i, x_j)^e)$ defined on an odd-gcd-closed (resp. odd-lcm-closed) set $S$ divides the $n\times n$ power LCM matrix $([x_i, x_j]^e)$ defined on $S$ in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for $n\le 3$ but they are both not true for $n\ge 4$.
In this paper, we will focus our attention on two basic semantic notions: linguistic expression and relation of synonymy (lexical synonymy). We will try to analyze the notions in three semantic fields: logic, linguistic, and pragmatic (F. Recanati), and see whether these three fields can provide us with satisfactory solutions. We will show that the pragmatic conception of semantics is not satisfactory in explaining neither the notion of linguistic expression nor the notion of synonymy - two very basic semantic notions; which could be done in logical and linguistic semantics. and Barbora Geistová Čakovská