Isotone mapping of free lattices

1968 ◽  
Vol 4 (4) ◽  
pp. 734-741 ◽  
Author(s):  
T. S. Fofanova
Keyword(s):  
Isotone Mapping

On Finitely Generated Simple Complemented Lattices

1981 ◽  
Vol 24 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Werner Poguntke
Keyword(s):  
Partially Ordered Sets ◽  
Partial Ordering ◽  
Isomorphic Copy ◽  
Ordered Sets ◽  
Finitely Generated ◽  
Partially Ordered ◽  
Isotone Mapping ◽  
Complemented Lattices ◽  
Element Set

Let L be a lattice, and let P and Q be partially ordered sets. We say that L is generated by P if there is an isotone mapping from P into L with its image generating L. P contains Q if there is a subset Q’ of P which, with the partial ordering inherited from P, gives an isomorphic copy of Q. For an integer n > 0, the lattice of partitions of an n-element set will be denoted by II(n); it is well-known that II(rc) is simple and complemented (cf. P. Crawley-R. P. Dilworth [1; p. 96]).

Common Fixed Points of Commuting Monotone Mappings

1967 ◽  
Vol 19 ◽  
pp. 617-620 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Points ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Common Fixed Points ◽  
Monotone Mappings ◽  
Partially Ordered ◽  
Isotone Mapping ◽  
Chain Complete ◽  
Empty Subset

We are concerned here with the existence of fixed or common fixed points of commuting monotone self-mappings of a partially ordered set into itself. Let X be a partially ordered set. A self-mapping ƒ of X into itself is called an isotone mapping if x ⩾ y implies ƒ(x) ⩾ ƒ(y). Similarly, a self-mapping ƒ of X into itself is called an antitone mapping if x ⩾ y implies ƒ(x) ⩽ ƒ(y). An element X0 ∈ X is called well-ordered complete if every well-ordered subset with x0 as its first element has a supremum. An element x0 ∈ X is called chain-complete if every non-empty chain C ⊆ X such that x ⩾ x0 for all x ∈ C, has a supremum. X is called a well-ordered-complete semi-lattice if every non-empty well-ordered subset has a supremum. X is called a complete semi-lattice if every non-empty subset of X has a supremum.

Isotone extensions and complete lattices

2019 ◽  
Vol 16 (4) ◽  
pp. 514-535
Author(s):  
Oleksiy Dovgoshey
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Lattices ◽  
Isotone Mapping ◽  
Necessary And Sufficient

The set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an isotone extension to an isotone mapping from X to Y is found.

Galois Connections and Pair Algebras

1969 ◽  
Vol 21 ◽  
pp. 498-501 ◽  
Author(s):  
J. C. Derderian
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Order Relation ◽  
Galois Connection ◽  
Ordered Sets ◽  
Residuated Mapping ◽  
Partially Ordered ◽  
Isotone Mapping ◽  
Order Relations

Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” .An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that(RM 1) xϕψ ≧ x for all x i n P;(RM 2) yψϕ ≦ for all y in Q.Let Q* denote the partially ordered set with order relation dual to that of Q.(A) The following conditions are equivalent:(i) ϕ: P → Q* is a Galois connection;(ii) ϕ: P → Q is a residuated mapping;(iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ.Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.

Some characterizations of pseudo-chains in pseudo-ordered sets

2020 ◽  
pp. 2150074
Author(s):  
Shashirekha B. Rai ◽  
S. Parameshwara Bhatta
Keyword(s):  
Fixed Point ◽  
Chain Condition ◽  
Fixed Point Property ◽  
Minimum Condition ◽  
Theoretic Approach ◽  
Point Property ◽  
Property A ◽  
Isotone Mapping ◽  
Complete Trellis

In this paper, we have proved that minimum condition in a pseudo-ordered set (psoset) is equivalent to the descending pseudo-chain condition. Characterization of a pseudo-chain in an acyclic psoset is obtained using graph theoretic approach as transitivity need not hold in psosets. Skala proved that every complete trellis has Fixed Point Property. A counterexample given in this paper shows that trellis having Fixed Point Property need not be complete. The notion of weakly isotone mapping and Strong Fixed Point Property is introduced in a psoset and the characterization of weakly isotone mapping is obtained. It is proved that a connected psoset containing a nontrivial cycle does not have Strong Fixed Point Property.

scholarly journals THE BEST SOLUTION FOR INEQUALITIES OF A O CROSS X LOWER THAN X FROM B O DOT X USING HIGH MATRIX RESIDUATION OF IDEMPOTENT SEMIRING

2017 ◽  
Vol 5 (1) ◽  
pp. 43
Author(s):  
Eka Susilowati ◽  
Ari Suparwanto
Keyword(s):  
Fixed Point ◽  
Complete Lattice ◽  
Binary Operation ◽  
Idempotent Semiring ◽  
Standard Definition ◽  
Isotone Mapping ◽  
Idempotent Semirings ◽  
Residuation Theory ◽  
High Matrix ◽  
Definition Of

Abstract A complete idempotent semiring has a structure which is called a complete lattice. Because of the same structure as the complete lattice then inequality of the complete idempotent semiring can be solved a solution by using residuation theory. One of the inequality which is explained is  where matrices A,X,B with entries in the complete idempotent semiring S. Furthermore, introduced dual product , i.e. binary operation endowed in a complete idempotent semirings S and not included in the standard definition of complete idempotent semirings. A solution of inequality  can be solved by using residuation theory. Because of the guarantee that for each isotone mapping in complete lattice always has a fixed point, then is also exist in a complete idempotent semirings. This of the characteristics is used in order to obtain the greatest solution of inequality . Keywords: complete lattice, complete idempotent semiring, dual Kleene Star, dual product, residuation theory

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