Homomonomerek

Homomonomerek is a coined term found in limited or introductory discussions that blend ideas from homomorphisms and monomials. It is not a standard, widely adopted concept in mainstream algebra, but is used in some didactic or exploratory texts to discuss how monomial-like objects behave under structure-preserving maps.

In informal terms, a homomonomerek refers to a monomial (or a well-defined set of monomials) that is preserved in a controlled way by a given family of algebra homomorphisms. Concretely, if f is an algebra homomorphism between polynomial-like structures, a monomial m is said to belong to a homomonomerek if f(m) is again a monomial (often up to multiplication by a unit in the target ring). This captures the idea of stability of monomial forms under maps that respect the underlying algebraic structure.

A typical simple example arises in a polynomial ring k[x1, ..., xn]. If one considers endomorphisms defined by sending each variable to a monomial (for instance, xi maps to xi or to xi^2), then every monomial maps to another monomial. In such a setting, the whole set of monomials can be viewed as a homomonomerek with respect to the chosen endomorphism. More refined versions may restrict to subsets of monomials whose exponent patterns are preserved under the endomorphisms.

Possible uses include illustrating invariants under homomorphisms, studying how monomial bases transform under structure-preserving maps, and supporting teaching examples in algebraic combinatorics. Because the term is not standardized, readers often encounter varying definitions, so context from the source is essential. See also homomorphism, monomial, polynomial ring, and invariant theory.