For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace $ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $<W>$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace
(1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace $. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied.
In an algebraic frame $L$ the dimension, $\dim (L)$, is defined, as in classical ideal theory, to be the maximum of the lengths $n$ of chains of primes $p_0<p_1<\cdots <p_n$, if such a maximum exists, and $\infty $ otherwise. A notion of “dominance” is then defined among the compact elements of $L$, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of $L$, including the frames $dL$ and $zL$ of $d$-elements and $z$-elements, respectively. The more concrete illustrations regarding the frame convex $\ell $-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if $A$ is a commutative semiprime $f$-ring with finite $\ell $-dimension then $A$ must be hyperarchimedean. The $d$-dimension of an $\ell $-group is invariant under formation of direct products, whereas $\ell $-dimension is not. $r$-dimension of a commutative semiprime $f$-ring is either 0 or infinite, but this fails if nilpotent elements are present. $sp$-dimension coincides with classical Krull dimension in commutative semiprime $f$-rings with bounded inversion.
For a vertex v of a connected oriented graph D and an ordered set W = [wi, w2,.. •, wk} of vertices of D, Ihe (directed distcince) representation of v with respect to W is the ordered fc-tuple r(v \ W) = (d(v, w1),d(v, w2 ), •.. ,d(v, wk )), where d(v, wi ) is Ihe directed distance from v to wi . The set W is a resolving set for D if every two distinct vertices of D have distinct representations. The minimum cardinality of a resolving set for D is the (directed distance) dimension dhn(D) of D. The dimension of a connected oriented graph need not be defined. Those oriented graphs with dimension 1 are characterized. We discuss the problem of determining the largest dimension of an oriented graph with a fixed order. It is shown that if the outdegree of every vertex of a connected oriented graph D of order n is at least 2 and dim(D) is defined, then dim(D) ≤ n - 3 and this bound is sharp.
For an ordered set W = {w1, w2,...,wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W)=(d(v,w1), d(v, w2),...,d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations. A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). For a basis W of G, a subset S of W is called a forcing subset of W if W is the unique basis containing S. The forcing number fG(W, dim) of W in G is the minimum cardinality of a forcing subset for W, while the forcing dimension f(G, dim) of G is the smallest forcing number among all bases of G. The forcing dimensions of some well-known graphs are determined. It is shown that for all integers a, b with 0 ≤ a ≤ b and b ≥1, there exists a nontrivial connected graph G with f(G) = a and dim(G) = b if and only if {a, b} ≠ {0, 1}.