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DEMONSTRATIO MATHEMATICAVol. X L INo 32008I. Chajda, M. Kolarik, H. LängerCHARACTERIZATIONS OF POSETS VIA W E A K STATESA b s t r a c t . Weak states on posets are defined which are in some analogy to stateson orthomodular posets used in axiomatic quantum mechanics. It is shown how certainproperties of the set of weak states characterize certain properties of the underlying poset.Orthomodular posets serve as algebraic models for logics in axiomaticquantum mechanics. States on them are considered which reflect the properties of states of the corresponding physical system. A crucial property ofsuch states is monotonicity. In analogy to these states we define so-calledweak states on an arbitrary poset. These weak states are also monotonousand play some role in the characterization of certain algebraic models ofquantum systems (cf. [2]). We use properties of the set of weak states inorder to characterize certain properties of the underlying poset. In this context semilattices play an important role. For the theory of semilattices werefer the reader to the recent monograph [1],In the following let V = (P, < ) be an arbitrary but fixed non-emptyposet.DEFINITION1. We call V trivialn a n d a\,...,anU(ai,...,an)6 P:=if |P| = 1.p u t L(ai,...,an){x
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 2008
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