By Peng Wei, Sun Bingnan, Tang Jinchun

In keeping with analytical equations, a catenary point is gifted for thefinite point research of cable constructions. in comparison with frequently used aspect (3-node point, 5-node element), a software with the proposed point is of lesscomputer time and higher accuracy.

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P (T ) ⊆ θ(Ult(B(T ))). Upper Semi-lattice Algebras and Combinatorics 33 Now, by 1. and 2. p (T ) ∪ {∅}. Moreover, by 1. we have ∅ = θu0 ∈ θ(Ult(B(T ))). p (T ) ∪ {∅}. 5. Let u ∈ Ult(B(T )) and set I = θu. 1. θu = ∅. By the same argument as in case i) 5. take t1 , . . , tn ∈ I, s1 , . . , sm ∈ / I and set / J}. p (T ) ∪ {∅} : t1 , . . , tn ∈ J and s1 , . . 2. θu = ∅. Let s1 , . . , sm ∈ T . p (T ) ∪ {∅} : s1 , . . , sm ∈ / −1 (V(I)), where s is again the J}. Thus, b = m j=1 −bsj = 0 and u ∈ s(b) ⊆ θ Stone representation mapping.

So, V = {F ∈ F(S) : ai ∈ F, and bj ∈ / F for i = 1, . . , n; j = 1, . . , m}, with ai ∈ F0 ∩ G0 , and bj ∈ / F0 ∩ G0 for i = 1, . . , n; j = 1, . . , m. Hence, ϕ−1 (V ) = {(F, G) ∈ F(S) × F(S) : (ai , bj ) ∈ (F ∩ G) × (F (S) \ (F ∩ G)) for i = 1, . . , n; j = 1, . . , m} = {(F, G) ∈ F(S) × F(S) : ai ∈ F ∩ G for i = 1, . . , n} / F ∩ G for j = 1, . . , m} ∩ {(F, G) ∈ F(S) × F(S) : bj ∈ = ({F ∈ F(S) : ai ∈ F for i = 1, . . , n}× {G ∈ F(S) : ai ∈ G for i = 1, . . , n}) ∩ ∩m / F } × F(S)) j=1 [({F ∈ F(S) : bj ∈ ∪ (F (S) × {G ∈ F(S) : bj ∈ / G})] Upper Semi-lattice Algebras and Combinatorics 41 which is clearly an open set containing (F0 , G0 ).

1, p. 255] for treealgebras. For t ∈ T and any finite anti-chain A(t) ⊆ (T, ≤) above t, e(t, A(t)) denotes bt · − s∈A(t) bs . Let b ∈ B(T ) \ {0}. Recall that b has a unique decomposition of the form (+): b = e(t1 , A(t1 )) + · · · + e(tn , A(tn )) / A(tj ) for all i, j where {e(ti , A(ti )) : 1 ≤ i ≤ n} are pairwise disjoint and ti ∈ distinct; see [1]. Let: supp+ (b) := {t1 , . . , tn } supp− (b) := A(t1 ) ∪ · · · ∪ A(tn ). Set µ+ (b) := |supp+ (b)| and µ+ (0) = 0. Note that the results of this section, appear in [16], extending the work done in [15].

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