tümomorfi

Tümomorfi is a term used in abstract algebra, specifically in the study of universal algebra and category theory. It refers to a type of homomorphism between universal algebras or, more generally, between objects in a category that are equipped with algebraic structures. A tümomorfi preserves all the operations defined on the algebras. In the context of universal algebra, if you have two algebras A and B and a function f from the universe of A to the universe of B, f is a tümomorfi if for every operation $\omega$ of arity n in the signature of the algebras, and for all elements $a_1, a_2, ..., a_n$ in the universe of A, the following holds: $f(\omega(a_1, ..., a_n)) = \omega(f(a_1), ..., f(a_n))$. This means the operation can be applied either before or after mapping the elements, and the result will be the same. The concept is crucial for understanding the structural relationships between different algebraic systems and for developing general theorems that apply across a wide range of algebraic structures. In category theory, a tümomorfi is often equivalent to an algebra homomorphism in the category of algebras of a given type.