Representations of monoid elements

Using place forms data structures may have different representations with respect to different monoids, for example, with respect to or the false value is equal to nil and the true value is a collection with a single element. Exclusive disjunction uses the same representation except its single generator is an involution rather then an idempotent element:

{false {}
 true {1 1}}

Logical conjunction uses the negation of logical disjunction because these are the two elements of the boolean algebra with two distinct elements. Integers may be represented as the multiplicity of a single generator in an additive monoid, but in a multiplicative monoid they are a multiset with prime numbers as keys and exponents as values:

{1 12}
{2 2, 3 1}

Polynomials also have their on additive and multiplicative representations based upon factorization. The essential point here is that there may be many different representations of a single data structure depending upon the monoid that it is currently being handled by and these representations can be described by place forms.