Numbrijoonils

Numbrijoonils are a class of abstract algebraic structures defined to explore the interaction between additive-like and multiplicative-like operations with an embedded numerical signature. A numbrijoonil consists of a finite set N together with two binary operations, denoted ⊕ and ⊗, and a function sig: N → Z, all defined with respect to a fixed modulus m > 1. The structure satisfies that (N, ⊕) forms a commutative monoid with identity 0, and (N, ⊗) forms a commutative semigroup with identity 1. Distributivity holds: a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c). The signature function is compatible with the operations in the sense that sig(a ⊕ b) ≡ sig(a) + sig(b) (mod m) and sig(a ⊗ b) ≡ sig(a) sig(b) (mod m). Elements of N may be interpreted as encoding both an additive component and a multiplicative component via their sig value.

In construction terms, numbrijoonils generalize rings by dropping the requirement that every element have an additive inverse, while retaining a distributive law and a numerical signature. The simplest nontrivial model occurs when N is finite and sig maps into Z/mZ; one can take N to be {0,1,...,k−1} with standard representatives and define ⊕ and ⊗ by modular addition and multiplication, respectively, complemented by a modulus-k signature.

Numbrijoonils are primarily of interest in abstract algebra and model theory, where they serve as test cases for theories that blend numerical invariants with algebraic operations. They have potential applications in coding theory and combinatorial enumeration, where the signature function provides a compact invariant.