Unary operations on completely distributive complete lattices

Since almost fifty years, Shannon entropy measure has been used as a very powerful tool when quantifying and processing the amount of randomness contained in probability distributions. In this paper we propose a lattice-valued entropy measure H ascribing to each lattice-valued possibilistic distribution π the value H(π) defined as the expected value (in the sense of lattice-valued Sugeno integral with infimum in the role of t-norm) of certain nonincreasing function of the values ascribed to the elements of the basic space by the possibilistic distribution π in question. The main result reads that, for completely distributive complete lattices, the entropy value ascribed to possibilistically independent product of a finite number of lattice-valued possibilistic distributions is defined by the supremum of the entropy values ascribed to particular distributions.

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Direct decompositions of completely distributive complete lattices

Czechoslovak Mathematical Journal ◽

10.21136/cmj.1955.100164 ◽

1955 ◽

Vol 05 (4) ◽

pp. 488-491

Author(s):

Ján Jakubík

Keyword(s):

Complete Lattices ◽

Completely Distributive ◽

Direct Decompositions

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Abstract reflexive sublattices and completely distributive collapsibility

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032226 ◽

1998 ◽

Vol 58 (2) ◽

pp. 245-260 ◽

Cited By ~ 12

Author(s):

W. E. Longstaff ◽

J. B. Nation ◽

Oreste Panaia

Keyword(s):

Banach Space ◽

Operator Theory ◽

Operator Algebras ◽

Galois Connection ◽

General Situation ◽

Subspace Lattice ◽

Complete Lattices ◽

Completely Distributive ◽

Subspace Lattices

There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice ℒ is completely distributive, then ℒ is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on ℒ such that ℒ/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

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