On interconnection between the numbers of Helly, Radon, and Caratheodory in the lattices with balanced frame (Q1333664)

Various approaches are known in lattice theory for defining independent subsets of elements. We recall them: Helly independence (\(h\)- independence), Radon independence (\(r\)-independence), and Goldie independence (\(g\)-independence). The supremum of the powers of independent subsets (with regard to the respective notions of independence) is called Helly number \(h(L)\), Radon number \(r(L)\), or Goldie number \(g(L)\) of a lattice \(L\), respectively. Furthermore, \(L\) will be a lattice with zero. A frame (from below) of a lattice \(L\) be a subset \(F\) of \(L\) such that for each non-zero element \(a\in L\) there exists a non-zero element \(b\in F\) with \(b\leq a\). Given the frame \(F\) of a lattice \(L\), let there exist the least integer \(k\geq 0\) satisfying the condition: if \(x\in F\), \(S\subseteq F\), \(| S|< \infty\), \(x\leq \lor S\), then \(x\leq\lor T\) for some subset \(T\subseteq S\), we have \(| T|\leq k\). This number \(k\) is called the Caratheodory number \(c_ F(L)\) of the lattice \(L\) by (with respect to) the frame \(F\). If such a \(k\) does not exist, we put \(c_ F(L)= \infty\). A subset \(A\subseteq L\backslash\{0\}\) is said to be \(g\)-independent if the following relation holds: \(a\land(\lor(A_ 1\backslash\{a\}))= 0\) for every element \(a\in A_ 1\), where \(A_ 1\) is an arbitrary finite subset of \(A\). A frame \(F\) of a lattice \(L\) is said to be balanced if for every finite \(g\)- independent subset \(X\subseteq F\backslash\{0\}\) and any elements \(x_ 1,x_ 2\in F\backslash\{0\}\) satisfying the condition \(x_ i\land (\lor X)= 0\) \((i=1,2)\), the relation (1) \(x_ 1\land (x_ 2\lor(\lor X))= 0\) holds if and only if the relation (2) \(x_ 2\land(x_ 1\lor(\lor X))= 0\) is satisfied. If \(L=\text{Co}(\mathbb{R}^{n-1})\) is the lattice of all convex subsets of \((n-1)\)-dimensional real Euclidean space, \(F\) is the totality of the atoms of \(L\), then the Helly number \(h\), the Radon number \(r\), and the Caratheodory number \(c_ F\) of the lattice \(L\) by the frame \(F\) satisfy the classical relation (3) \(h= r= c_ F= n\). In the present article we establish a relation of type (3), where the natural number \(n\) is the length of some naturally ordered subset of a lattice \(L\), for lattices with balanced frame. Furthermore, we establish for these lattices the relation (4) \(h= r= g= c_ F= n\), where \(g\) is the Goldie number.