Partially ordered ring

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a partially ordered ring is a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> (A, +, ·), together with a compatible partial order, that is, a <a href="/facts/Partial_order/qb4a3FAX">partial order</a> 
 
 
 
 
 ≤
 
 
 
 {\displaystyle \,\leq \,}
 
 on the <a href="/facts/Underlying_set/TkuXca24">underlying set</a> A that is compatible with the ring operations in the sense that it satisfies:

x
        ≤
        y
        
           implies 
        
        x
        +
        z
        ≤
        y
        +
        z
      
    
    {\displaystyle x\leq y{\text{ implies }}x+z\leq y+z}

and

0
        ≤
        x
        
           and 
        
        0
        ≤
        y
        
           imply that 
        
        0
        ≤
        x
        ⋅
        y
      
    
    {\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y}

for all 
 
 
 
 x
 ,
 y
 ,
 z
 ∈
 A
 
 
 {\displaystyle x,y,z\in A}
 
. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring 
 
 
 
 (
 A
 ,
 ≤
 )
 
 
 {\displaystyle (A,\leq )}
 
 where 
 
 
 
 A
 
 
 {\displaystyle A}
 
's partially ordered additive <a href="/facts/Partially_ordered_group/xoLNpEqH">group</a> is <a href="/facts/Archimedean_group/tzEyqUcR">Archimedean</a>.
An ordered ring, also called a totally ordered ring, is a partially ordered ring 
 
 
 
 (
 A
 ,
 ≤
 )
 
 
 {\displaystyle (A,\leq )}
 
 where 
 
 
 
 
 ≤
 
 
 
 {\displaystyle \,\leq \,}
 
 is additionally a <a href="/facts/Total_order/xnTtmuYJ">total order</a>.
An l-ring, or lattice-ordered ring, is a partially ordered ring 
 
 
 
 (
 A
 ,
 ≤
 )
 
 
 {\displaystyle (A,\leq )}
 
 where 
 
 
 
 
 ≤
 
 
 
 {\displaystyle \,\leq \,}
 
 is additionally a <a href="/facts/Lattice_order/5ak6ufky">lattice order</a>.

Partially ordered ring open-in-new

Partially ordered ring