“Esakia space”的意思、由来-开放百科全书

词条

Esakia space

释义

Definition
Equivalent definitions
Esakia morphisms
Notes
References

In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974.^[1] Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality --- the dual equivalence between the category of Heyting algebras and the category of Esakia spaces.

Definition

For a partially ordered set {{math|(X,≤)}} and for {{math|x∈ X}}, let {{math|↓x {{=}} {y∈ X : y≤ x}}} and let {{math|↑x {{=}} {y∈ X : x≤ y} }}. Also, for {{math|A⊆ X}}, let {{math|↓A {{=}} {y∈ X : y ≤ x for some x∈ A}}} and {{math|↑A {{=}} {y∈ X : y≥ x for some x∈ A} }}.

An Esakia space is a Priestley space {{math|(X,τ,≤)}} such that for each clopen subset {{math|C}} of the topological space {{math|(X,τ)}}, the set {{math|↓C}} is also clopen.

Equivalent definitions

There are several equivalent ways to define Esakia spaces.

Theorem:^[2] The following conditions are equivalent:

(i) {{math|(X,τ,≤)}} is an Esakia space.

(ii) {{math|↑x}} is closed for each {{math|x∈ X}} and {{math|↓C}} is clopen for each clopen {{math|C⊆ X}}.

(iii) {{math|↓x}} is closed for each {{math|x∈ X}} and {{math|↑cl(A) {{=}} cl(↑A)}} for each {{math|A⊆ X}} (where {{math|cl}} denotes the closure in {{math|X}}).

(iv) {{math|↓x}} is closed for each {{math|x∈ X}}, the least closed set containing an up-set is an up-set, and the least up-set containing a closed set is closed.

Esakia morphisms

Let {{math|(X,≤)}} and {{math|(Y,≤)}} be partially ordered sets and let {{math|f : X → Y}} be an order-preserving map. The map {{math|f}} is a bounded morphism (also known as p-morphism) if for each {{math|x∈ X}} and {{math|y∈ Y}}, if {{math|f(x)≤ y}}, then there exists {{math|z∈ X}} such that {{math|x≤ z}} and {{math|f(z) {{=}} y}}.

Theorem:^[3] The following conditions are equivalent:

(1) {{math|f}} is a bounded morphism.

(2) {{math|f(↑x) {{=}} ↑f(x)}} for each {{math|x∈ X}}.

(3) {{math|f⁻¹(↓y) {{=}} ↓f⁻¹(y)}} for each {{math|y∈ Y}}.

Let {{math|(X, τ, ≤)}} and {{math|(Y, τ′, ≤)}} be Esakia spaces and let {{math|f : X → Y}} be a map. The map {{math|f}} is called an Esakia morphism if {{math|f}} is a continuous bounded morphism.

Notes

1. ^Esakia (1974)
2. ^Esakia (1974), Esakia (1985).
3. ^Esakia (1974), Esakia (1985).

References

Esakia, L. (1974). Topological Kripke models. Soviet Math. Dokl., 15 147–151.
Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi.

1 : General topology

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。