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2012•Fixed Point Theory and ApplicationsOpen access

The fixed set of a derivation in lattices

Xiao Long Xin

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Abstract

The related properties of derivations in lattices are investigated. We show that the set of all isotone derivations in a distributive lattice can form a distributive lattice. Moreover, we introduce the fixed set of derivations in lattices and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices. MSC: 06B35, 06B99.

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The related properties of derivations in lattices are investigated. We show that the set of all isotone derivations in a distributive lattice can form a distributive lattice. Moreover, we introduce the fixed set of derivations in lattices and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices. MSC: 06B35, 06B99.

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Available abstract

The related properties of derivations in lattices are investigated. We show that the set of all isotone derivations in a distributive lattice can form a distributive lattice. Moreover, we introduce the fixed set of derivations in lattices and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices. MSC: 06B35, 06B99.

Key concepts: Distributive lattice, Complete lattice, Mathematics, Distributive property, Lattice (music), Map of lattices, Congruence lattice problem, Fixed point

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