ALGEBRAS WITH ACTIONS AND AUTOMATA

.


INTRODUCTION (ALGEBRAIC FUNCTORS)
The following paper is directed to specialists. Customary shorthand theorems, and arguments are freely used.
Triple theory (cf. [], [2]VI, [3]3) showed that the notion of "algebraic structures" should be viewed as a relative notion: structures of a given species are algebraic over some other category, and this is best described in terms of the forgetful functor from some category K of "algebras" into a "base" category B.
We say U is tripleable or monadic (cf. references above).
We will use only the axiomatic properties of monadic functors.
We will use as our basic notion the following slightly stronger one: . U is of finite rank if (iv) U creates filtered colimits.
1.3 REMARK. For B Sets, "monadic" and "algebraic" are the same, again by [4], 3.3, since in Sets coequalizers are retractions. As we will see in section 2, "usual" algebraic structures, i.e. those with operations of finite arity (in particular L a w v e r e type algebras) have algebraic forgetful functors of finite rank, if B has enough properties of Sets.
We prefer "algebraic" to "monadic" because of the following quite important structure theorems and because generation of congruences, isomorphism theorems, Z a s s e n h a u s lemma, J o r d a n- By P a r 's monadicity criterion we need to show (for the first part of our assertion) that U creates absolute coequalizers.
But since V preserves absolute coequalizers (as such, any functor does), U creates them.
Second assertion: It is easy to verify that a functor (e.g. V,W) into a category having limits,colimits of a given kind which creates these limits, colimits respectively, preserves them. This proves the second part of the assertion" U creates "things" because V preserves them, W creates them, and U maps them into the original things, because V creates (uniqueness). U right adjoint holds by the first assertion since the lifting theorem provides all coequalizers in K via W. q.e.d.
Each time a (full) subcategory is equationally defined, we will use the following theorem for proving the inclusion functor to be algebraic.
The lower triangle commutes, since f extends f; the frame commutes, since f is compatible with any w il'''in----i in Y.
An induction on I shows the left triangle to commute. Hence fi is necessarily induced out of the coproduct by the Aw o fi.
. jn This f extends f (take w=id.) and is a l-homomorphism as is shown by i using the "coproduct-row" and commutativity of (I), the commutative upper rectangle in (2), and the recursive definition of A. This proves (i). (v) is proved similarly using the fact that a binary functor which preserves filtered colimits in each place, commutes with them, see [5], 6.5.
Let us now turn to many-sorted algebraic theories and their models.

ThI I I
We draw now some conclusions from this theorem which will be needed for the proof of the main theorem below. (ii) follows from 1.5 and FZ/G being a coequalizer of pr Pr2 G-'-FK relative V; for the homomorphisms cf. proof of (i).

EXTENSION OF MODELS
(iii) follows from the fact that each T Th I is generated by its arrows with codomain in I, since any arrow is induced by those.   7)). An analysis of the proofs shows that the theorem is still valid for countable T (i.e. generated by a countable Z) and the category of countable sets, since B xstill preserves colimits although it has no right adjoint. Similarly: R,S-bimodules and group operations for R a group.

ALGEBRAS WITH ACTIONS
The examples at the end of section 2 become more interesting for the applications, if the "scalar" or "input" components are fixed, cf.
[7] and the examples in 3.4 and 3.5. We show here that under certain conditions (3. 3.2) fixing components still gives monadic or algebraic forgetful functors. 3.

ALGEBRAS WITH ACTIONS
A theory with J-action (J E Sets) is an I 0 J-sorted theory T having the following factorization property: Every T-arrow AB factorization AB pr A' (A,A' E J*, B I *) has a Now let T] Thj be the full subcategory of T with object set J* (Example: Tj theory of unitary rings) and R Mod(Tj,B)_ be a fixed Tj-model over some base category _B with (given) finite products.
Then a T-model M with MiT R is called an R-model (for T).

UR'
The lemma follows now from R' R' and by considering the J-J components of (I) (existence of an extension of (fj)jj) and by the observation that any f F(Rj,Bi)ITj-----R' in Mod (Tj,) extends to an f F(Rj,Bi)-----R' in Mod(T,B) (uniqueness of the extension) q.e.d. for the heterogeneous case. The same is true for theorem 3.3 because of its "relative" formulation and since we did not use the universal properties of in the proof.
A list of [symmetric] monoidal categories which are "nice" in the above sense is to be found in [15]:3.|. Theorem 3.3 then gives in particular the following examples of action-categories with