C-semiring

In abstract algebra, a c-semiring (that is, a constraint-based semiring) is a tuple <"A",+,X,0,1> such that:

• "A" is a set and 0, 1 are elements of "A".
• + is the additive operation and is a commutative (i.e., +("a","b") = +("b","a")) and associative (i.e., +("a",+("b","c")) = +(+("a","b"),"c")) operation such that +("a",0) = "a" = +(0,"a") (i.e., 0 is its unit element).
• + is defined over (possibly infinite) sets of elements of "A" as follows:
  • for all "a" which are elements of "A", +({"a"}) = "a";
  • +(empty set) = 0 and +("A") = 1;
  • +(U"A_i", "i" element of "S") = +({+("A_i"), "i" element of "S"}) for all sets of indices of "S" (flattening property).
• X is called the multiplicative operation, is a binary, associative and commutative operation such that 1 is its unit element and "a" X 0 = 0 = 0 X "a" (i.e., 0 is its absorbing element);
• X distributes over +.