ON THE DIRECTLY AND SUBDIRECTLY IRREDUCIBLE MANY-SORTED ALGEBRAS

A theorem of general algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this paper we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff according to which every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true for the many-sorted algebras.


Introduction
Some theorems of ordinary universal algebra can not be automatically generalized to world of many-sorted universal algebra, see e.g., [3] and [4] for the case of a representation theorem of Birkhoff-Frink, or [5] for that one of the injectivity of the insertion of the generators in the relatively free many-sorted algebras.
Our main aim in this paper is to prove, in the third section, that, under a mild condition on the supports of the factors in the definition of the concept of directly reducible many-sorted algebra, every finite many-sorted algebra can also be represented as a product of a finite family of finite directly irreducible manysorted algebras.In addition, in the fourth section, for completeness, we show that the many-sorted counterpart of the well-known theorem of Birkhoff about the representation of every single-sorted algebra as a subdirect product of subdirectly irreducible single-sorted algebras, is also true for the many-sorted algebras.
In the second section we define those notions and constructions from the theory of many-sorted sets and algebras which are indispensable in order to attain the above indicated goals.
In this section we begin by defining for an arbitrary, but fixed, set of sorts S, those concepts of the theory of S-sorted sets which we need in order to state the notions of many-sorted signature, algebra, subalgebra, homomorphism from a many-sorted algebra to another, product of a family of many-sorted algebras, and congruence on a many-sorted algebra.Definition 1.Let S be a set of sorts.
(1) A word on S is a mapping w : n / / S, for some n ∈ N. We denote by S the underlying set of the free monoid on S, i.e., the set n∈N S n of all mappings from the finite ordinals to S.Moreover, we call the unique mapping λ : ∅ / / S, the empty word on S. (2) An S-sorted set A is a function (A s ) s∈S from S to U , where U is a Grothendieck universe, fixed once and for all, and the support of A, denoted by supp(A), is the set For every set of sorts S, the support of an S-sorted set A is a subset of S, hence it really a mapping supp : U S / / Sub(S).In the following proposition we gather together some useful properties of the mapping supp.Proposition 1.Let S be a set of sorts, A, B be two S-sorted sets, (A i ) i∈I a family of S-sorted sets, and Φ an S-sorted equivalence on an A. Then the following properties hold: (1) Following this we define the concepts of many-sorted signature, algebra, and homomorphism.
Definition 2. A many-sorted signature is a pair (S, Σ), where S is a set of sorts and Σ an S-sorted signature, i.e., a function from S × S to U which sends a pair (w, s) ∈ S × S to the set Σ w,s of the formal operations of arity w, sort (or coarity) s, and rank (or biarity) (w, s).Sometimes we will write σ : w / / s to indicate that the formal operation σ belongs to Σ w,s .From now on, to shorten notation, we will write Σ instead of (S, Σ).Definition 3. Let Σ be a many-sorted signature.Then (1) The S × S-sorted set of the finitary operations on an S-sorted set A, denoted by HOp S (A), is where A w = i∈|w| A w i , with |w| denoting the length of the word w.
where, for (w, s) ∈ S × S, F w,s is a mapping from Σ w,s to Hom(A w , A s ).
For a pair (w, s) ∈ S × S and a formal operation σ ∈ Σ w,s , in order to simplify the notation, the operation from A w to A s corresponding to σ under F w,s will be written as F σ instead of F w,s (σ).(3) A Σ-algebra is a pair (A, F ), abbreviated to A, where A is an S-sorted set and F a structure of Σ-algebra on A. (4) A Σ-homomorphism from A to B, where B = (B, G), is a triple (A, f, B), abbreviated to f : A / / B, where f is an S-sorted mapping from A to B such that, for every (w, s) ∈ S × S, σ ∈ Σ w,s , and We denote by Alg(Σ) the category of Σ-algebras.
Sometimes, to avoid any confusion, we will denote the structures of Σ-algebra of the Σ-algebras A, B, . . ., by F A , F B , . . ., respectively, and the components of F A , F B , . . ., as F A σ , F B σ , . . ., respectively.Next we define the concept of subalgebra of a many-sorted algebra.Definition 4. Let A be a Σ-algebra and X ⊆ A.
(1) Let σ be such that σ : w / / s, i.e., a formal operation in Σ w,s .We say that X is closed under the operation We say that X is a subalgebra of A if X is closed under the operations of A. We denote by Sub(A) the set of all subalgebras of A.
Following this we recall the concept of product of a family of many-sorted algebras.
Definition 5. Let (A i ) i∈I be a family of Σ-algebras, where, for i ∈ I, (1) The product of (A i ) i∈I , denoted by i∈I A i , is the Σ-algebra ( i∈I A i , F ) where, for every σ : w / / s in Σ, F σ is defined as The i-th canonical projection, pr i , is the homomorphism from i∈I A i to A i defined, for every s ∈ S, as follows We define next the concept of subfinal many-sorted algebra, since it will be used in the following section in an essential way.Definition 6.A Σ-algebra A is subfinal if, for every Σ-algebra B, there is at most a homomorphism from B to A.
We point out that the subfinal many-sorted algebras are subobjects of the final many-sorted algebra, therefore their underlying many-sorted sets are subfinal.
We define now the concepts of many-sorted congruence on a many-sorted algebra and of many-sorted quotient algebra of a many-sorted algebra modulo a manysorted congruence.
Definition 7. Let A be a Σ-algebra and Φ an S-sorted equivalence on A. We say that Φ is an S-sorted congruence on A if, for every (w, s) ∈ (S − {λ}) × S, σ : w / / s, and a, b ∈ A w we have that We denote by Cgr(A) the set of S-sorted congruences on A and by Cgr(A) the ordered set (Cgr(A), ⊆).Definition 8. Let A be a Σ-algebra and Φ ∈ Cgr(A).The many-sorted quotient algebra of A modulus Φ, A/Φ, is the Σ-algebra ((A s /Φ s ) s∈S , F ) where, for every σ : w / / s in Σ, the operation

Directly irreducible many-sorted algebras.
In this section we show that every finite many-sorted algebra is isomorphic to a finite product of finite directly irreducible many-sorted algebras.
Unlike that which happens for single-sorted algebras, there exists subfinal, but not final, many-sorted algebras that are isomorphic to products of nonempty families of nonsubfinal many-sorted algebras, and this is so because the supports of the factors can strictly contain the support of the product.This suggest that in the definition of directly reducible many-sorted algebra we should require that the supports of the factors of the product be included in the support of the many-sorted algebra under consideration.This additional condition will allow us to obtain the theorem about the representation of a finite many-sorted algebra as a product of a finite family of finite directly irreducible many-sorted algebras.Definition 9. Let A be a Σ-algebra.We say that A is directly reducible if A is isomorphic to a product of two nonsubfinal Σ-algebras such that their supports are included in that of A. If A is not directly reducible, then we will say that A is directly irreducible.
Obviously, every subfinal Σ-algebra is directly irreducible.Moreover, every finite Σ-algebra A such that, for some S ∈ S, card(A s ) is a prime number is also directly irreducible.
As for single-sorted algebras, we define the factorial congruences on a manysorted algebra, from which we will obtain a characterization of the directly irreducible many-sorted algebras.
Definition 10.Let Φ and Ψ be two congruences on a Σ-algebra A. We say that Φ and Ψ are a pair of factorial congruences on A if they satisfy the following conditions: Proposition 2. Let A and B be two Σ-algebras.Then the kernels of the canonical projections from A×B to A and B, denoted by Ker(pr 0 ) and Ker(pr 1 ), respectively, are a pair of factorial congruences on A × B. Proposition 3. If Φ and Ψ is a pair of factorial congruences on A, then we have that A ∼ = A/Φ × A/Ψ.

Proof. Let f : A
/ / A/Φ × A/Ψ be the S-sorted mapping defined, for every s ∈ S and a ∈ A s∈S , as and f is surjective.Proposition 4. Let A be a Σ-algebra.Then A is directly irreducible if and only if ∆ A and ∇ A is the only pair of factorial congruences on A. Theorem 1.Every finite Σ-algebra is isomorphic to a product of a finite family of finite directly irreducible Σ-algebras.
Proof.Let A be a finite Σ-algebra.If card( s∈S A s ) = 0, then A is irreducible.Let A be such that card( s∈S A s ) = n + 1, with n ≥ 0, and let us assume the theorem for every finite Σ-algebra B such that card( s∈S B s∈S ) ≤ n.If A is directly irreducible, then we are finished.Otherwise, we have that A ∼ = A 0 × A 1 , with A 0 and A 1 nonsubfinal Σ-algebras and such that, for i = 0, 1, supp(A i ) ⊆ supp(A).
Let A i T be, for i = 0, 1 and T = supp(A) = supp(A 0 )∩supp(A 1 ), the Σ-algebra (A i T, F A i T ), where A i T , for every s ∈ S, is defined as and F A i T is defined, for every (w, s) ∈ S × S, as where α As is the unique mapping from ∅ to A s .The definition of the many-sorted structure is sound since, for σ : w / / s, both Im(w) ⊆ T and s ∈ T can not occur.
From this it follows that A ∼ = A 0 T × A 1 T and, for i = 0, 1, that card(A i T ) < card(A), hence, by the induction hypothesis, we can assert that where, for j ∈ p and h ∈ q, B j and C k are directly irreducible.Therefore, Remark.If for a Σ-algebra A the lattice (Cgr(A) − {∆ A }, ⊆) has a minimum Φ, then the lattice Cgr(A) has the form: the maximum congruence on A. The congruence Φ, called the monolith of A and denoted by M A , has the property that M A = Cg A (δ s, (a,b) ), for every s ∈ S and every (a, b) ∈ M A s , with a = b, where δ s, (a,b) is the S-sorted set which has as s-th coordinate the set {(a, b)} and as t-th coordinate, for t = s, the empty set, and Cg A the generated congruence operator for A.
We define next the simple many-sorted algebras, that are a special kind of subdirectly irreducible algebra.Definition 13.Let A be a Σ-algebra.We say that A is simple if A is subfinal or Cgr(A) has exactly two congruences.Moreover, we say that a congruence Φ on A is maximal if the interval [Φ, ∇ A ] in the lattice Cgr(A) has exactly two congruences.
As for single-sorted algebras, also many-sorted algebras it is true that the quotient many-sorted algebra of a many-sorted algebra by a congruence on it is simple if and only if the congruence is maximal or the congruence is the maximum congruence on the many-sorted algebra.
Proposition 7. Let A be a Σ-algebra and Φ a congruence on A. Then A/Φ is simple if and only if Φ is a maximal congruence on A or Φ = ∇ A .
In the following proposition we gather together some relations between the simple, the subdirectly irreducible, and the directly irreducible many-sorted algebras.
Proposition 8. Every simple many-sorted algebra is subdirectly irreducible and every subdirectly irreducible many-sorted algebra is directly irreducible.
We prove next, as was announced in the introduction of this paper, the manysorted counterpart of the well-known theorem of Birkhoff about the representation of every single-sorted algebra as a subdirect product of subdirectly irreducible single-sorted algebras, is also true for the many-sorted algebras.
Theorem 2 (Birkhoff).Every many-sorted algebra is isomorphic to a subdirect product of a family of subdirectly irreducible many-sorted algebras.
Corollary 2. Every finite many-sorted algebra is isomorphic to a subdirect product of a finite family of finite subdirectly irreducible many-sorted algebras.
) i∈I of S-sorted sets, we denote by i∈I A i the S-sorted set such that, for every s ∈ S, is nonempty, by i∈I A i the S-sorted set such that, for every s ∈ S, is finite and, for every s ∈ supp(A), A s is finite, or, what is equivalent, if A is finite.If A and B are S-sorted sets, then A ⊆ B if, for every s ∈ S, A s ⊆ B s and A ⊆ fin B if A is finite and A ⊆ B.Moreover, we denote by Sub(A) the set of all S-sorted sets X such that, for every s ∈ S, X s ⊆ A s .Finally, given a set I and an I-indexed family (A i