Split partial and total order of rings (#61)

Rings/Orders/Partial/Definition.agda

@@ -0,0 +1,25 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open Ring R
+open Group additiveGroup
+open Setoid S
+
+record PartiallyOrderedRing {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
+  field
+    orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
+    orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
+  open SetoidPartialOrder pOrder