Algebraic and Arithmetic Lattices . Part I 1
نویسنده
چکیده
(1) Let L be a non empty reflexive transitive relational structure and x, y be elements of L. If x ≤ y, then compactbelow(x)⊆ compactbelow(y). (2) For every non empty reflexive relational structure L and for every element x of L holds compactbelow(x) is a subset of CompactSublatt(L). (3) For every relational structure L and for every relational substructure S of L holds every subset of S is a subset of L. (4) For every non empty reflexive transitive relational structure L with l.u.b.’s holds the carrier of L is an ideal of L. (5) Let L1 be a lower-bounded non empty reflexive antisymmetric relational structure and L2 be a non empty reflexive antisymmetric relational structure. Suppose the relational structure of L1 = the relational structure of L2 and L1 is up-complete. Then the carrier of CompactSublatt(L1) = the carrier of CompactSublatt(L2).
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