Semiring

<p>In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a semiring is an <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a>. Semirings are a generalization of <a href="/facts/Ring_(algebra)/JnCUXz63">rings</a>, dropping the requirement that each element must have an <a href="/facts/Additive_inverse/n8gKUmCf">additive inverse</a>. At the same time, semirings are a generalization of <a href="/facts/Lattice_(order)/5ak6ufky">bounded</a> <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattices</a>.
</p><p>The smallest semiring that is not a ring is the <a href="/facts/Two-element_Boolean_algebra/8nIWUpwd">two-element Boolean algebra</a>, for instance with <a href="/facts/Logical_disjunction/jbI2EGlM">logical disjunction</a> 
  
    
      
        ∨
      
    
    {\displaystyle \lor }
  
 as addition. A motivating example that is neither a ring nor a lattice is the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a> 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
 (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the <a href="/facts/Function_composition/r5Pvnmth">function composition</a> of <a href="/facts/Endomorphism/s7NNlSea">endomorphisms</a> over any <a href="/facts/Commutative_monoid/6AHve7p8">commutative monoid</a>.
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Semiring open-in-new

Semiring