Generically Ordinary Morphisms in Algebra

• Generically ordinary morphisms are defined as families of mapping formulas that generalize classical homomorphisms to provide a unified approach for single-sorted and many-sorted algebras.
• The framework extends Fujiwara's basic mapping-formulas and derivors by introducing polyderivors, which enable flexible transformations and compatibility across diverse algebraic systems.
• Categorical and 2-categorical structures underpin the theory, establishing robust equivalence relations and transformation mechanisms within generalized algebraic theories.

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 (Vidal et al., 2012).

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 (Vidal et al., 2012).

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 (Vidal et al., 2012).