An isomorphism theorem for anti-ordered sets
Daniel Abraham Romano2008, Filomat
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AI-generated Abstract
This paper presents a new isomorphism theorem for anti-ordered sets within the framework of constructive mathematics. It explores the properties of apartness and relations related to anti-orders and quasi-anti-orders, ultimately establishing conditions under which certain mappings between ordered sets can be characterized as isomorphic.
Key takeaways
- (5) ( [18], Theorem 5) Let q be a coequality relation on a semigroup S with apartness.
- where c(R) is the biggest quasi-antiorder relation on X under R = α∩Coker(ϕ), q = c(R) ∪ c(R) −1 and γ is the antiorder induced by the quasi-antiorder c(R).
- In that case, in the definition of quasi-antiorder relation on the ordered set (X, =, =, α) under the antiorder α , we must add the following condition (∀x, y ∈ X)((x, y) ∈ σ =⇒ ¬((y, x) ∈ σ)),
- Then q = σ ∪ σ −1 is a coequality relation on X such that (X/q, = 1 , = 1 ) is an ordered set under the antiorder relation β defined by (xq, yq) ∈ β ⇐⇒ (x, y) ∈ σ.
- Let (X, =, =, α), (Y, =, =, β) be ordered sets under antiorders α and β respective, and let ϕ : X −→ Y be a strongly extensional mapping from X into Y .
