The second isomorphism theorem on ordered set ... - Kjm.pmf.kg.ac.rs

237f is called reve**rs**e isot**on**e if and **on**ly if(∀x, y ∈ S)((f(x), f(y)) ∈ β ⇒ (x, y) ∈ α).**on**gly extensi**on**al mapping f is called an **on**to, isot**on**e and reve**rs**e isot**on**e. X and Y called isomorphic, in symbolX ∼ = Y , if exists an **on** τ ⊆ X/ × X is a quasi-antiorder **on** X if and **on**ly ifτ ⊆ α(⊆≠), τ ⊆ τ ∗ τ, τ ∩ τ −1 = ∅.**rs**t propositi**on** gives some informati**on** **on** quasi-antiorder:Lemma 1. If τ is a quasi-antiorder **on** X, then the relati**on** q = τ ∪τ −1 is a coequalityrelati**on** **on** X.Proof. (1) q = τ ∪ τ −1 ⊆≠ ∪ ≠ −1 =≠ ∪ ≠=≠ (because ≠ −1 =≠).(2) q = τ ∪ τ −1 = τ −1 ∪ τ = (τ ∪ τ −1 ) −1 = q −1 .(3) q = τ ∪τ −1 ⊆ (τ ∗τ)∪(τ −1 ∗τ −1 ) ⊆ (τ ∗τ)∪(τ ∗τ −1 )∪(τ −1 ∗τ)∪(τ −1 ∗τ −1 ) =τ ∗ (τ ∪ τ −1 ) ∪ τ −1 ∗ (τ ∪ τ −1 ) = (τ ∪ τ −1 ) ∗ (τ ∪ τ −1 ) = q ∗ q. □Now we give the **on**d**ordered** **set** under antiorderα and let τ be a quasi-antiorder **on** X.Lemma 2. Let τ be a quasi-antiorder relati**on** **on** X, q = τ ∪ τ −1 . **on**θ **on** X/q, defined by(aq, bq) ∈ θ ⇔ (a, b) ∈ q,is an antiorder **on** X/q. **on**gly extensi**on**al surjectivereve**rs**e isot**on**e mapping and holds θ ◦ π = τ, in which case θ is equal to τ ◦ π −1 .Proof. We shall check **on**ly linearity of the relati**on** θ: Let aq and bq be arbitraryelements of **set** X/q such that aq ≠ bq. **on** we need a lemma in which we will give explanati**on** abouttwo coequalities α and β **on** a **set** (X, =, ≠) such that β ⊆ α.

238Lemma 3. ([5]) Let α and β be coequality relati**on**s **on** a **set** X with apartness suchthat β ⊂ α. **on** β/α **on** X/α, defined by β/α = {(xα, yα) ∈ X/q×X/q :(x, y) ∈ β}, is an coequality relati**on** **on** X/α and there exists the str**on**gly extensi**on**aland embedding bijecti**on**f : (X/α)/(β/α) → (X/β).So, at the end of this article, we are in positi**on** to give a descripti**on** of c**on**necti**on**between quasi-antiorder ρ and quasi-antiorder σ of **set** X with apartness such thatσ ⊆ ρ.**set**, ρ and σ quasi-antiorde**rs** **on** X such that σ ⊆ ρ.**on** σ/ρ defined byσ/ρ = {(x(ρ ∪ ρ −1 ), y(ρ ∪ ρ −1 )) ∈ X/(ρ ∪ ρ −1 ) × X/(ρ ∪ ρ −1 ) : (x, y) ∈ σ},is a quasi-antiorder **on** X/(ρ ∪ ρ −1 ) and(X/(ρ ∪ ρ −1 )/((σ/ρ) ∪ (σ/ρ) −1 )) ∼ = X/(σ ∪ σ −1 )holds.Proof.(1) Put q = (ρ ∪ ρ −1 ), and c**on**struct the f**ac**tor-**set** (X/q, = q , ≠ q ). If a and b areelements of X, then(aq, bq) ∈ σ/ρ ⇔ (a, b) ∈ σ⇒ (a, b) ∈ ρ (because σ ⊆ ρ)⇒ (a, b) ∈ q⇔ aq ≠ q bq.(aq, cq) ∈ σ/ρ ⇔ (a, c) ∈ σ⇒ (∀b ∈ X)((a, b) ∈ σ ∨ (b, c) ∈ σ)⇔ (∀bq ∈ X/q)((aq, bq) ∈ X/q ∨ (bq, cq) ∈ q).Suppose that (σ/ρ) ∩ (σ/ρ) −1 ≠ ∅, and let (aq, bq) ∈ (σ/ρ) ∩ (σ/ρ) −1 .

240**ordered** **set** under an antiorder α, σ a quasiantiorder**on** X such that σ ∩ σ −1 = ∅. Let A = {τ : τ is quasi-antiorder **on** X suchthat τ ⊆ σ}. Let B be the family of all quasi-antiorder **on** X/q, where q = σ ∪ σ −1 .For τ ∈ A, we define a relati**on**ψ(τ) = {(aq, bq) ∈ X/q × X/q : (a, b) ∈ τ}.**on**gly extensti**on**al, injective and surjective mappingfrom A **on**to B and for τ, µ ∈ A we have τ ⊆ µ if and **on**ly if ψ(τ) ⊆ ψ(µ).Proof.(1) f is well-defined functi**on**:Let τ ∈ A. **on** X/q by Lemma 3.Let τ, µ ∈ A with τ = µ. If (aq, bq) ∈ ψ(τ) then (a, b) ∈ τ = µ, so (aq, bq) ∈ψ(µ). Similarly, ψ(µ) ⊆ ψ(τ). **on**.(2) ψ is an injecti**on**. Let τ, µ ∈ A, ψ(τ) = q ψ(µ). Let (a, b) ∈ τ. Since (aq, bq) ∈ψ(τ) = q ψ(µ), we have (a, b) ∈ µ. Similarly, we c**on**clude µ ⊆ τ. So, µ = τ.(3) ψ is str**on**gly extensi**on**al. Let τ, µ ∈ A, ψ(τ) ≠ q ψ(µ) i. e. let there exists anelement (aq, bq) ∈ ψ(τ) and (aq, bq)#ψ(µ). **on**clude τ ≠ µ.(4) ψ is **on**to. Let M ∈ B. We define a relati**on** µ **on** X as follows:µ = {(x, y) ∈ X × X : (xq, yq) ∈ M}.µ is a quasi-antiorder. In f**ac**t:

242[4] D. A. Romano,Semivaluati**on** **on** Heyting Field, Kragujev**ac** Journal of Mathematics,20 (1998), 24–40[5] D. A. Romano, A **on** Subdirect Product of Semigroups with Apartness,Filomat, 14 (2000), 1–8[6] A. S. Troelstra, D. van Dalen, C**on**structivism in Mathematics, An Introducti**on**,Volume II, North-Holland, Amsterdam (1988).