---
title: Generically Ordinary Morphisms in Algebra
url: https://www.emergentmind.com/topics/generically-ordinary-morphisms
type: topic
---

# Generically Ordinary Morphisms in Algebra

Generically ordinary morphisms generalize the classical concept of homomorphisms between algebras by introducing families of mapping formulas, allowing for a more flexible and expressive morphism framework. Originating with Fujiwara's work in the single-sorted setting, these morphisms have been extended to many-sorted algebras via the concept of polyderivors, subsuming standard signature morphisms and generalizing earlier frameworks such as derivors (Goguen–Thatcher–Wagner) and basic mapping-formulas (Fujiwara). The theory establishes a unified approach to morphisms, transformations, and equivalence relations in universal algebra, with categorical and 2-categorical structures connecting to the theory of institutions and 2-institutions [1201.5239].

## 1. Single-Sorted Foundations and Basic Mapping-Formulas

For single-sorted signatures, a signature $\Sigma$ is defined as $\Sigma = \bigcup_{n \in \mathbb{N}} \Sigma_n$, with each $\sigma \in \Sigma_n$ possessing arity $n$. Given two $\Sigma$-algebras $A = (A, (F_\sigma)_{\sigma \in \Sigma})$ and $B = (B, (G_\sigma)_{\sigma \in \Sigma})$, a family of basic mapping-formulas $\Phi$ from $A$ to $B$—termed a Fujiwara morphism—consists of:
- A finite set $\Phi = \{\phi_\mu \mid \mu \in P\}$ of mapping variables.
- For each $n \in \mathbb{N}$ and $\sigma \in \Sigma_n$, a term $P^n_{(\phi_\mu, \sigma)}$ in the term algebra $T_\Sigma(\Phi \times \{v_0, ..., v_{n-1}\})$.

These families induce a $\Sigma$-algebra structure on $B^\Phi$, with the operations constructed via evaluation of mapping formulas, as specified in the term:

\[
F_\sigma^{B^\Phi}(b_0, ..., b_{n-1}) = (P^n_{\phi_\mu, \sigma}^{B}(b_0, ..., b_{n-1}))_{\mu \in P} \in B^\Phi
\]

This structure underpins the notion of generically ordinary morphisms in the single-sorted case.

## 2. Conjugation and Transformations Between Morphisms

A key innovation is the introduction of an equivalence relation called conjugation on families of basic mapping-formulas. Given two morphisms $(\Phi, P)$ and $(\Psi, Q)$, and a set of $\Sigma$-equations $\mathcal{H}$, a transformation

\[
L: (\Phi, P) \longrightarrow (\Psi, Q)
\]

is an assignment $L_\nu \in T_\Sigma(\Phi)$ (for each $\nu \in \Psi$) such that for all $n$ and $\sigma \in \Sigma_n$:

\[
Q^n_{\psi_\nu,\sigma}(L^{\sharp}(P^n_{\phi_\mu,\sigma})_{\mu \in \Phi}) \equiv_{\mathcal{H}} L^{\sharp}_\nu((P^n_{\phi_\mu,\sigma})_{\mu \in \Phi})
\]

Two morphisms are conjugate modulo $\mathcal{H}$ if there exist invertible transformations in both directions. Conjugation corresponds exactly to inner self-isomorphisms of the derived-operation algebra $B^\Phi$: conjugate morphisms define the same subalgebra of $\text{End}(B^\Phi)$ up to isomorphism.

## 3. Polyderivors and the Many-Sorted Generalization

The framework is generalized to many-sorted signatures, $\Sigma = (S, F)$, with $S$ a set of sorts and $F = (F_{w, s})_{(w, s) \in S^* \times S}$ the operation symbols, each of arity $w$ (a word in $S^*$) and co-arity $s \in S$. A polyderivor $d: \Sigma \to \Lambda$ is defined by:
- A sort-map $\varphi: S \to T^*(T)$, where $\varphi(s)$ is a word of target sorts.
- For each $(w, s) \in S^* \times S$, a derived-term assignment

\[
\theta_{w, s}: \Sigma_{w, s} \to T_\Lambda(X_{\text{ext}(\varphi)(w)})_{\varphi(s)}
\]

where $\text{ext}(\varphi): S^* \to T^{**}$ is the monoid extension. The derived terms must satisfy arity-compatibility equations for projections and tuplings.

Polyderivors unify standard signature morphisms, Fujiwara basic mapping-formulas, and Goguen–Thatcher–Wagner derivors, providing a single categorical structure for ordinary and generalized morphisms.

## 4. Categorical Structure: Composition, 2-Categories, and 2-Institutions

The category of many-sorted signatures and polyderivors admits a composition operation, establishing a category isomorphic to the Kleisli category for a monad on the standard category of many-sorted signatures. Introducing the notion of transformation between polyderivors provides a 2-category structure on the category of many-sorted signatures and polyderivors.

This yields a derived 2-category of many-sorted specifications, in which the equivalence of many-sorted specifications of Hall and Bénabou is proven. The 2-categorical methods extend the abstraction provided by the institution theory of Goguen and Burstall, introducing the concept of a 2-institution, a strict generalization that supports the invariance of generalized many-sorted terms under polyderivors and compatibility with their transformations [1201.5239].

## 5. Equivalence of Categories and Applications to Algebraic Theories

A significant consequence is the demonstration of the equivalence between the categories of many-sorted clones (Hall) and algebraic theories (Bénabou), via the established categorical framework. Enabou algebras serve as the algebraic counterpart of the finitary many-sorted algebraic theories of Bénabou, and the categorical constructions ensure the realization of generalized many-sorted terms in algebras is invariant under polyderivors and compatible with their transformations.

## 6. Relation to Prior Frameworks and Generalization

Polyderivors strictly encompass earlier frameworks, notably unifying:
- Standard signature morphisms,
- Basic mapping-formulas (Fujiwara morphisms),
- Derivors (Goguen–Thatcher–Wagner).

By subsuming these, the structure of generically ordinary morphisms offers a canonical setting for expressing and analyzing morphisms and transformations across single-sorted and many-sorted algebraic structures. The result is a robust base for the categorical study of algebraic theories, signature morphisms, and the equivalence of expressive frameworks, providing deep connections to the ongoing development of universal algebra and institution theory [1201.5239].

Source: https://www.emergentmind.com/topics/generically-ordinary-morphisms