We will consider some other choices in Sections 4.3-4.4 below.
for generality, we conclude this section by stating the precise technical conditions needed to construct E P,R and E P,R,S,T : • P is a semiring • R is a P -module (e.g, a vector space), meaning that it comes equipped with an associative and commutative addition operation with an identity element 0, and also a multiplication operation As a matter of notation, note that above and in Tables 1-2, we overload "+" to denote any of the addition operations within P, R, S, T ; overload "0" to denote their respective additive identities; and overload concatenation to denote any of the multiplication operations within or between The second and third rows define the operations between two elements p1, r1, s1, t1 and p2, r2, s2, t2 , while the last two rows define the identities.