I've been trying to find a rig-like structure (a set R with a monoid structure (R,⋅,1) and a (commutative) monoid structure (R,+,0) such that the multiplication distributes over addition) in which 0 is not an absorber with respect to multiplication, i.e. 0⋅r≠0 for some r∈R.It should be possible to construct such a structure since you have to require absorption as a separate axiom in a rig.So far I haven't been successful as I don't have any experience with rigs but maybe someone else has a neat idea?

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I&#039;ve been trying to find a rig-like structure (a set R with a monoid structure (R,⋅,1) and a (commutative) monoid structure (R,+,0) such that the multiplication distributes over addition) in which 0 is not an absorber with respect to multiplication, i.e. 0⋅r≠0 for some r∈R.It should be possible to construct such a structure since you have to require absorption as a separate axiom in a rig.So far I haven&#039;t been successful as I don&#039;t have any experience with rigs but maybe someone else has a neat idea?

yotaniwc · Accepted Answer

Step 1  Let (R,⋅,1) be a commutative monoïd such that  r2=r  for all  r∈R  (for example, you can start with any Boolean ring (R,+,⋅,0,1). Concrete example are given, by (Z2Z)I for any nonempty set I, for example)  Then set +=⋅ and 0=1.  Hence (R,+,0)=(R,⋅,1) is a commutative monoïd. Moreover, for all r,s,t∈R, we have  r⋅(s+t)=r(st)=rst and  r⋅s+r⋅t=(rs)(rt)=r2st=rst,  so you have the distributivity property. Now   r⋅0=r⋅1=r  for all r∈R.

hysgubwyri3 · Accepted Answer

Step 1  Let E be the set of intervals of the real line of the form [a,b] where a≤0 and b≥0.   Define  [a,b]⊕[c,d]=[M∈(a,c),Max(b,d)]  and  [a,b]⊗[c,d]=[a+c,b+d].  One can check that (E,⊕) and (E,⊗) are abelian monoids, both having identity [0,0], and furthermore ⊗ distributes over ⊕.  If it had an absorbing zero element, it would be a full-fledged semiring... but it obviously does not, since the ⊕ identity is the same as the ⊗ identity, it is obviously not absorbing.

I've been trying to find a rig-like structure (a set R with a monoid structure (R,\cdot,1) and

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