CARTESIAN CLOSED ALGEBRAIC CATEGORIES Vijaya L. Gompa

Home , Exponential object, Subcategory

International Journal of Pure and Applied Mathematics Volume 78 No. 2 2012, 215-231 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu

CARTESIAN CLOSED ALGEBRAIC CATEGORIES

Vijaya L. Gompa Professor and Chair Department of Mathematics Troy University-Dothan Campus P.O. Box 8368, Dothan, Alabama, 36304, USA

Abstract: It is well known that the category Alg(1) of all universal algebras with one unary operation is cartesian closed. We extend this result for the category Alg(Σ) of universal algebras of a ﬁxed type Σ all of whose operations are unary. That is, we show that the category Alg(Σ) is cartesian closed. In fact, we characterize some cartesian closed full isomorphism closed subcategories of Alg(Σ) using a concept called T -friendly.

AMS Subject Classiﬁcation: 08A25, 08A60, 08A30, 08C05, 17A30, 18A40, 18B99, 18D15, 54A05 Key Words: topological category, topological functors, universal algebra, topological algebra, cartesian closed, canonical function spaces

1. Introduction

The category T op of topological spaces and continuous maps is observed to be inadequate (for example, products and quotients in T op do not commute) and is replaced by some convenient topological categories which happen to be cartesian closed (see [4], [16]). An extensive study has been made on carte-

c 2012 Academic Publications, Ltd. Received: March 13, 2012 url: www.acadpubl.eu 216 V.L. Gompa sian closed topological categories (see, for example, [4], [16], [1], [3], [9], [10], [11], [12], [13], [14], and [15]), in particular, some characterizations are obtained for cartesian closedness of topological categories (see [9]) and monotopological categories (see [13]). Cartesian closed topological categories are especially im- portant in the theory of topological algebra for the construction of free algebras over topological objects and, obviously, for the construction of quotients (see [14] and [15]). The present author studied cartesian closedness of topologically algebraic categories of universal algebras. It turned out that the category of universal algebras has to be cartesian closed whenever the associated category of topological algebras is cartesian closed (see [8]). This created an interest in finding cartesian closed categories of universal algebras. Even though, there has been an ample work found in the literature on monoidal closed categories of universal algebras (see, for example, [7] and [6]), it seems a little has been done about cartesian closed algebraic categories. It is well known that the category Alg(1) of all universal algebras with one unary operation is cartesian closed (see, [2] page 408). We extend this result for the category Alg(Σ) of universal algebras of a fixed type Σ all of whose operations are unary. That is, we show that the category Alg(Σ) is cartesian closed. In fact, we characterize some cartesian closed full isomorphism closed subcategories of Alg(Σ) using a concept called T -friendly. It turns out that a full isomorphism closed subcategory A of Alg(Σ) containing a free A-object T on a singleton is cartesian closed iff the set homA(A × T,B) of all A-morphisms from A×T to B is an A-object with a certain Σ-structure for any two A-objects A and B. We will also describe full isomorphism closed subcategories of the category Alg(Ω) of universal algebras of a fixed type Ω that have canonical function spaces. It turns out that any object A in a category of universal algebras with one n-ary operation (n is any fixed positive integer) with canonical function spaces has a simple algebraic structure. Namely, there exist n universal algebras A1,...,An such that n-ary operation on Ai is the i-th canonical projection n Ai → Ai (1 ≤ i ≤ n) and A is isomorphic to the product A1 ×···× An.

2. Preliminaries

A family Ω = (nj)j∈J of natural numbers indexed by some set J is called a type. The index set J is called the order of Ω. In the following, we let a type n Ω = (nj)j∈J be ﬁxed. A pair (A, (ωj)j∈J ) of a set A and a family ωj : A j → A CARTESIANCLOSEDALGEBRAICCATEGORIES 217

(j ∈ J) of mappings is called an Ω-algebra (see, for example, [5]). For the sake of simplicity, we write A instead of of the pair (A, (ωj)j∈J ) and ωj,A for the nj- ary operation ωj on A. If the Ω-algebra A is clear from the context, we drop the suffix A in denoting its nj-ary (j ∈ J) operation. Moreover, the symbol Σ is used instead of Ω in case nj = 1 for each j ∈ J. A mapping f : A → B between two Ω-algebras A and B is said to be an Ω-homomorphism iff for n n n n each j ∈ J, f ◦ωj,A = ωj,B ◦f where n = nj and f : A → B is the mapping with the obvious definition (a1, . . . , an) → (fa1, . . . , fan). The symbol Alg(Ω) denotes the category whose objects are Ω-algebras and whose morphisms are Ω-homomorphisms. A category A is called cartesian closed iff it has finite products and for each A-object A the functor (A ×−) : A → A is co-adjoint (see, for example, [2] page 407). Equivalently, a category with finite products is cartesian closed iff for any two A-objects A and B there exists an object BA and a A-morphism ev : A × BA → B such that for each A-morphism f : A × C → B there exists a unique A-morphism f¯ : C → BA so that the diagram

A × C ❅ id × f¯ ❅ f A ❅ ❄ ❅❘ A ✲ A × B ev B

A (idA is the identity map on A) commutes. B is called a power object and f¯ : C → BA is called the exponential morphism for f : A × C → B. A construct (i.e., a concrete category over Set) A is said to have canonical function spaces iff A has finite concrete products, A is cartesian closed, and ev : A × BA −→ B can be chosen such a way that the underlying set for the power object BA is the set of all A-morphisms from A to B and ev is the restriction of the canonical evaluation map in Set. Equivalently, a construct A has canonical function spaces iff A has finite concrete products, A is cartesian closed, and each constant function between A-objects is an A-morphism (see [2], page 415). In this work, we assume that all subcategories are full and isomorphism closed. The fact that the most of the natural subcategories fall into this class justifies our assumption. 218 V.L. Gompa

3. Cartesian Closed Subcategories of Alg(Σ)

In the following discussion, we reserve the symbol T for a Σ-algebra generated by a distinguished member 0 in T (i.e., for any t ∈ T there exist j1 ∈ J,...,jn ∈ J such that ωjn,T ◦ . . . ◦ ωj1,T (0) = t), equipped with a family (δj)j∈J of Σ- homomorphisms δj : T → T satisfying

δj(0)= ωj,T (0). (3.0)

A Σ-algebra A is said to be T-friendly iﬀ for each t ∈ T there exists a function At : A → A, such that A0 = idA, (3.1)

ωj,A ◦ At = Aωj,T (t), (3.2) and

At ◦ ωj,A = Aδj (t) (3.3) for all j ∈ J and for all t ∈ T . A subcategory A of Alg(Σ) is called T-friendly iff each A-object is T - friendly. Consider the following examples. Example 3.1. Equip the set N of nonnegative integers with the unary operation ω : N → N that maps any integer n to its successor n + 1 and take δ = ω. (The distinguished member of N is the integer zero.) For any object A n in Alg(1) with a unary operation u, define An := u to be the composition of u to itself n times. It is straight forward to see that Alg(1) is N-friendly. Example 3.2. Let T be the set of all n-tuples (n ≥ 1) of members of J together with a distinguished point 0. For each j ∈ J, define

ωj(0) = (j),ωj (j1,...,jn) = (j1,...,jn, j),

δj(0) = (j), and δj(j1,...,jn) = (j, jn,...,j1).

Then T is a Σ-algebra with the unary operations ωj (T is the Σ-word algebra on the alphabet {0}) and δj’s are Σ-homomorphisms. For any Σ-algebra A, deﬁne A0 := idA and

A(j1,...,jn) = ωjn,A ◦ . . . ◦ ωj1,A. With these deﬁnitions, Alg(Σ) is T -friendly. CARTESIANCLOSEDALGEBRAICCATEGORIES 219

Example 3.3. If (J, +, 0) is a monoid, then the subcategory A of Alg(Σ) consisting of all J-sets (A Σ-algebra A is said to be a J-set iﬀ ω0,A = idA and ′ ωj,A ◦ ωj′,A = ωj+j′,A for any j ∈ J, j ∈ J.) is J-friendly: Indeed, the unary ′ ′ ′ operations ωj,J(j ∈ J) on J are given by ωj,J (j ) = j + j (j ∈ J) and the ′ ′ ′ homomorphisms δj (j ∈ J) on J are deﬁned by δj(j )= j + j (j ∈ J). For any A-object A and any j ∈ J, Aj is the nj-ary operation ωj,A on A. If A and B are any two Σ-algebras, then we write homA(A × T,B) for the Σ-algebra whose underlying set is the set of all Σ-homomorphisms from A × T to B and whose nj-ary (unary) operation ωj assigns the Σ-homomorphism f ◦ (idA × δj) for any morphism f : A × T → B in homA(A × T,B) , i.e.,

ωjf = f ◦ (idA × δj). (3.4)

If A is T -friendly and a ∈ A, then deﬁne gA,a : T → A by

gA,a(t) := Ata (3.5) for any t ∈ T .

Lemma 3.4. If A is T -friendly, then gA,a is a Σ-homomorphism and

gA,ωj a = gA,a ◦ δj (3.6) for any a ∈ A and j ∈ J.

Proof. gA,a is a Σ-homomorphism by (3.2) and (3.6) follows from (3.3).

Proposition 3.5. Suppose A is a subcategory of Alg(Σ) closed under ﬁnite products such that A is T -friendly and homA(A × T,B) is an A-object for any two A-objects A and B, then A is cartesian closed. A The power object B associated with A and B is homA(A × T,B) . The exponential morphism f¯ : C → BA for the A-morphism f : A × C → B is given by

f¯(c) := f ◦ (idA × gC,c) (3.7) for any c ∈ C. The evaluation morphism ev : A × BA → B is deﬁned by

ev(a, f) := f(a, 0) (3.8) for any a ∈ A and f ∈ BA. 220 V.L. Gompa

Proof. It is enough to show that f¯ and ev are A-morphisms such that ev ◦ (idA × f¯ = f and that f¯ is unique with this property (see [2], pages 407, 408). f¯ is an A-morphism because, for any c ∈ C and j ∈ J, ¯ f(ωjc) = f ◦ (idA × gC,ωj c) by (3.7) = f ◦ (idA × gC,c ◦ δj) by (3.6)

= ωj[f ◦ (idA × gC,c)] by (3.4)

= ωj[f¯(c)] by (3.7). ev is an A-morphism:

ev[ωj(a, f)] = ev(ωja, ωjf)

= (ωjf)(ωja, 0) by (3.8)

= f(ωja, δj 0) by (3.4)

= f(ωja, ωj0) by (3.0)

= ωjf(a, 0) since f is an A-morphism

= ωj[ev(a, f)] by (3.8).

ev ◦ (idA × f¯)(a, c) = ev(a, f¯(c)) = f¯(c)(a, 0) by (3.8) = f(a, C0c) by (3.7) = f(a, c) by (3.1).

Thus ev ◦ (idA ◦ f¯) = f. It remains to show that f¯ is unique with this A property. Suppose g : C → B is any A-morphism such that ev ◦(idA ◦g)= f. Let c ∈ C, a ∈ A, and t ∈ T . Choose indices j1 ∈ J,...,jn ∈ J such that

ωjn ◦ . . . ◦ ωj1 (0)= t.

Using the fact that δj’s are Σ-homomorphisms agreeing with ωj’s at 0 (by (3.0)), we can see that

δj1 ◦ . . . ◦ δjn (0)= t.

g(ωjn ◦ . . . ◦ ωj1 (c))(a, 0) = [(ωjn ◦ . . . ◦ ωj1 )(g(c))](a, 0)

= g(c)(a, δj1 ◦ . . . ◦ δjn (0)) CARTESIANCLOSEDALGEBRAICCATEGORIES 221

= g(c)(a, t).

Thus

g(ωjn ◦ . . . ◦ ωj1 (c))(a, 0) = g(c)(a, t). Similarly, ¯ ¯ f(ωjn ◦ . . . ◦ ωj1 (c))(a, 0) = f(c)(a, t). However, the left hand side of the above two equalities is the same as ¯ f(a, ωjn ◦ . . . ◦ ωj1 (c)) because ev ◦ (idA × g) = f and ev ◦ (idA × f) = f. Thus f¯(c)(a, t)= g(c)(a, t). This means f¯ = g.

Corollary 3.6. Alg(Σ) is cartesian closed. If (J, +, 0) is a monoid, then the category of J-sets is cartesian closed.

Proof. This follows from the above proposition and the discussion in examples (3.2) and (3.3). Note that, if A and B are J-sets then homA(A × J, B) is a J-set because δj+j′ = δj′ ◦ δj.

We now establish the converse of Proposition (3.5). Namely: Proposition 3.7. Any cartesian closed subcategory A of Alg(Σ) containing an object T such that A is T -friendly contains homA(A × T,B) for any two A-objects A and B. Moreover, the power objects, the exponential morphisms, and the evaluation morphisms in A are described as in Proposition (3.5).

Proof. Suppose A and B are any two A-objects. Write D for the power object BA in A and let EV : A × D → B be the evaluation morphism in A. Deﬁne ϕ : D → homA(A × T,B) by

ϕ(d)= EV ◦ (idA × gD,d)

(gD,d, deﬁned in (3.5), is the exponential morphism for ϕ(d)) for any d ∈ D. ′ We show that ϕ is an isomorphism. If ϕ(d) = ϕ(d ), then gD,d = gD,d′ by the uniqueness of an exponential morphism. In particular, d = gD,d(0) = ′ gD,d′ (0)= d . Thus ϕ is one-one. ¯ If f ∈ homA(A × T,B), then there exists a unique A-morphism f : T → D such that EV ◦(idA ×f¯)= f. Writing f¯(0)= d, we see that f¯ = gD,d since these two A-morphisms agree at the generator 0 of T . Hence ϕ(d) = f. Therefore, ϕ is onto. Suppose d ∈ D and j ∈ J. Then

ϕ(ωjd) = EV ◦ (idA × gD,ωj d) 222 V.L. Gompa

= EV ◦ (idA × gD,d ◦ δj)hboxby(3.6)

= ωj[EV ◦ (idA × gD,d)] by (3.4)

= ωjϕ(d).

Thus ϕ is an isomorphism and clearly EV (a, d) = ev(a, ϕ(d)), where ev is defined as in (3.8). Since A is full isomorphism closed subcategory of Alg(Σ) , homA(A × T,B) can be taken as a power object associated with A and B, ev as the evaluation morphism and the exponential morphism f¯ : C → D for f : A × C → B such that EV ◦ (idA × f¯)= f may be replaced by ϕ ◦ f¯. Corollary 3.8. A subcategory A of Alg(Σ) closed under finite products containing an object T such that A is T -friendly is cartesian closed iff homA(A × T,B) is an A-object for any two A-objects A and B. Corollary 3.9. A subcategory A of Alg(Σ) closed under finite products containing a free A-object T on a singleton is cartesian closed iff homA(A × T,B) is an A-object for any two A-objects A and B.

Proof. It is suﬃcient to prove that A is T -friendly. Assume that T is a free A-object on the set {0}. Then clearly δj : T → T is the unique Σ- homomorphism that extends the association δj(0)= ωj,T (0) required by (3.0). Suppose A is an A-object. We show that A is T -friendly. For any a ∈ A, write fA,a for the unique Σ-homomorphism fA,a : T → A such that

fA,a(0)= a. (3.9)

For any t ∈ T , deﬁne At : A → A by

At(a)= fA,a(t) (3.10) for any a ∈ A. (3.1) is clear from (3.9). For any j ∈ J, t ∈ T , and a ∈ A,

Aωj,T (t)(a) = fA,a(ωj,T (t)) by (3.10) = ωj,A ◦ fA,a(t) since fA,a is a Σ-homomorphism

= ωj,A ◦ At(a) by (3.10), which implies that (3.2) is valid. To verify (3.3), ﬁrst note that

fA,ωj,A(a)(0) = ωj,A(a) by (3.9) CARTESIANCLOSEDALGEBRAICCATEGORIES 223

= ωj,A ◦ fA,a(0) by (3.9)

= fA,a ◦ ωj,T (0) since fA,a is a Σ-homomorphism

= fA,aoδj(0) by (3.0).

Since fA,ωj,A(a) and fA,a ◦ δj are Σ-homomorphisms agreeing at 0, we have

fA,ωj,A(a) = fA,a ◦ δj. (3.11)

Hence,

At ◦ ωj,A(a) = At(ωj,A(a))

= fA,ωj,A(a)(t) by (3.10) = fA,a ◦ δj(t) by (3.11)

= Aδj (t)(a) by (3.10).

Thus (3.3) is valid.

4. Subcategories of Alg(Ω) with Canonical Function Spaces

For any set A and for any positive integer n, dn,A denotes the map from A to An that assigns to each member a of A the n-tuple (a, . . . , a) each of whose co- ordinates are equal to a. The following proposition shows that any subcategory of Alg(Σ) with canonical function spaces is simply Set. Proposition 4.1. Suppose A is a subcategory of Alg(Ω) that is closed under ﬁnite products. Then (a) constant functions between any two A-objects are A-morphisms iﬀ for any A-object A and for any j ∈ J,

ωj,A ◦ dnj ,A = idA; (4.1)

(b) A has canonical function spaces iﬀ equation (4.1) holds and for any two A-objects A and B, the set H := homA(A, B) of all A-morphisms from A to B is an A-object such that

ωj,H(f1,...,fn)= ωj,B ◦ (f1 ×···× fn) ◦ dn,A, (4.2)

ωj,B[ωj,B(f1a1,...,f1an),...,ωj,B(fna1,...,fnan)] = ωj,B(f1a1,...,fnan) (4.3) 224 V.L. Gompa

where j ∈ J, n = nj, f1 ∈ H,...,fn ∈ H, a1 ∈ A, . . . , an ∈ A. A In this case exponential object B in A is homA(A, B); the exponential morphism f¯ : C → BA in A, associated with an A-morphism f : A × C → B, is deﬁned by f¯(c)(a)= f(a, c); (4.4) and the evaluation morphism ev : A × BA → B in A is a restriction of the canonical evaluation map in Set.

Proof. Note that a constant function f : B → A with the constant value n a ∈ A is an A-morphism iﬀ for any j ∈ J (n = nj), ωj,A ◦f = f ◦ωj,B, or, ωj,A ◦ dn,A(a) = a. Thus statement (a) is valid. Suppose A has canonical function spaces. Then clearly (4.1) holds since constant functions are A-morphisms. A is closed under homA functor by the deﬁnition. Assume that A and B are any two A-objects and let H := homA(A, B). If f1 ∈ H,...,fn ∈ H, and a ∈ A, then, since the restriction ev : A × H → B of the canonical evaluation map in Set is an evaluation morphism in A, we have

[ωj,H(f1,...,fn)](a) = ev[a, ωj,H (f1,...,fn)]

= ev[ωj,A(a, . . . , a),ωj,H (f1,...,fn)] by (4.1)

= ev[ωj,A×H((a, f1),..., (a, fn))]

= ωj,B[ev(a, f1), . . . , ev(a, fn)], since ev is an A − morphism,

= ωj,B(f1a, . . . , fna).

Thus (4.2) holds. Moreover, if a1 ∈ A, . . . , an ∈ A, then

ωj,B(f1a1,...,fnan) = ωj,B(ev(a1,f1), . . . , ev(an,fn))

= ev[ωj,A(a1, . . . , an),ωj,H (f1,...,fn)], since ev is an A − morphism,

= [ωj,H(f1,...,fn)](ωj,A(a1, . . . , an))

= ωj,B[f1(ωj,A(a1, . . . , an),...,fn(ωj,A(a1, . . . , an))] by (4.2)

= ωj,B(ωj,B(f1a1,...,f1an),...,ωj,B(fna1,...,fnan)).

Conversely, assume that A is closed under homA and satisﬁes (4.1) - (4.3). Assume A, B, and C are A-objects, and f : A × C → B is an A-morphism. CARTESIANCLOSEDALGEBRAICCATEGORIES 225

Write H := homA(A, B). Since A is closed under the formation of ﬁnite products, A × C has the obvious algebraic structure. Since the diagram A × C ❅ id × f¯ ❅ f A ❅ ❄ ❅❘ ✲ A × Hev B commutes (f¯ is deﬁned by (4.4)), it is enough to show that f¯ and ev are A- morphisms. Let j ∈ J, n = nj, and c1,...cn ∈ C. For any a ∈ A,

f¯(ωj,C(c1,...,cn))(a) = f(a, ωj,C(c1,...,cn)) by (4.4)

= f(ωj,A(a, . . . , a),ωj,C (c1,...,cn)) by (4.1)

= f(ωj,A×C((a, c1),..., (a, cn)))

= ωj,B(f(a, c1),...,f(a, cn))

= ωj,B(f¯(c1)a, . . . , f¯(cn)a)

= ωj,H(f¯(c1),..., f¯(cn))(a) by (4.2).

n Thus f¯ ◦ ωj,C = ωj,H ◦ f¯ . Hence f¯ is an A-morphism. Similarly, if a1, . . . , an ∈ A and f1,...,fn ∈ H, then

ev[ωj,A×H((a1,f1),..., (an,fn))] = ev[ωj,A(a1, . . . , an),ωj,H (f1,...,fn)]

= [ωj,H(f1,...,fn)](ωj,A(a1, . . . , an))

= ωj,B[f1(ωj,A(a1, . . . , an),...,fn(ωj,A(a1, . . . , an))] by (4.2)

= ωj,B(ωj,B(f1a1,...,f1an),...,ωj,B(fna1,...,fnan))

= ωj,B(f1a1,...,fnan) by (4.3)

= ωj,B(ev(a1,f1), . . . , ev(an,fn)). Thus ev is an A-morphism.

Remarks 4.2. If A has canonical function spaces, then for any two A- objects A and B, j ∈ J, n = nj, H := homA(A, B), and f1 ∈ H,...,fn ∈ H, the map ωj,B ◦ (f1 ×···× fn) ◦ dn,A, being equal to ωj,H(f1,...,fn), is an A-morphism. Consequently, for any i ∈ J, m = ni, a1 ∈ A, . . . , am ∈ A, we have ωj,B[ωi,B(f1a1,...,f1am),...,ωi,B(fna1,...,fnam)]

= ωi,B[ωj,B(f1a1,...,fna1),...,ωj,B(f1am,...,fnam)]. (4.5) 226 V.L. Gompa

5. A Structure Theorem

We conclude our work with a structure theorem for some categories of universal algebras with one operation. First we have a result as a consequence of Proposition (4.1). Lemma 5.1. Suppose A is a subcategory of Alg(Ω) closed under ﬁnite products, admitting free objects on sets with nj (j ∈ J) elements. Then A has canonical function spaces iﬀ A is closed under homA (i.e., for any two A-objects A and B, the set H := homA(A, B) of all A-morphisms from A to B is an A-object whose nj-ary (j ∈ J) operation is given by (4.2)), and for any A-object A, i ∈ J, j ∈ J, n = nj, m = ni, a ∈ A, ars ∈ A (1 ≤ r ≤ m, 1 ≤ s ≤ n), ωj(a, . . . , a)= a, (5.1)

ωj(ωi(a11, . . . , am1),...,ωi(a1n, . . . , amn))

= ωi(ωj(a11, . . . , a1n),...,ωj(am1, . . . , amn)), (5.2) and, for i = j,

ωj(ωj(a11, . . . , an1),...,ωj(a1n, . . . , ann)) = ωj(a11, . . . , ann). (5.3)

In particular, the subcategory of Alg(Ω) deﬁned by the above three identities has canonical function spaces.

Proof. Suppose A has canonical function spaces. Note that (5.1) is just a restatement of (4.1). Let i ∈ J, j ∈ J, m = ni, n = nj, A be an A- object, and ars ∈ A (1 ≤ r ≤ m, 1 ≤ s ≤ n). Let B be the free A-object on Nm := {1,...,m}. Let fs : B → A be the A-morphism that extends the association r → ars (r ∈ Nm) for 1 ≤ s ≤ n. With these A-morphisms equations (4.5) and (4.3) imply (5.2), and (5.3). Conversely, assume that A is closed under homA satisfying (5.1) - (5.3). For any two A-objects A and B, j ∈ J, n = nj, A-morphisms f1 : A → B,...,fn : A → B, members a1, . . . , an in A, write ars = fsar for 1 ≤ r, s ≤ n. Then (5.3) implies (4.3). Hence by Lemma (4.1), A has canonical function spaces. To prove the last statement, assume that A is the subcategory of Alg(Ω) deﬁned by the identities (5.1), (5.2), and (5.3). In order to show that A has canonical function spaces, it remains to prove that A is closed under homA functor. Suppose A and B are any two A-objects and write H := homA(A, B). First we show that H is closed under ωj,H (j ∈ J) deﬁned in (4.2). CARTESIANCLOSEDALGEBRAICCATEGORIES 227

Let n = nj, f1,...,fn ∈ H. Deﬁne g : A → B by

g(a)= ωj,B(f1a, . . . , fna) for any a ∈ A. For any i ∈ J, m = ni, a1 ∈ A, . . . , am ∈ A, g(ωi,A(a1, . . . , am)) = ωj,B[f1ωi,A(a1, . . . , am),...,fnωi,A(a1, . . . , am)]

= ωj,B[ωi,B(f1a1,...,f1am),...,ωi,B(fna1,...,fnam)]

= ωi,B[ωj,B(f1a1,...,fna1),...,ωj,B(f1am,...,fnam)] by (5.2)

= ωi,B(ga1, . . . , gam). Thus g ∈ H. Moreover, since (5.1) - (5.3) hold in B, they are also valid in H by the deﬁnition of operations on H. Hence H is an A-object.

The following example illustrates some objects of Alg(Ω) satisfying the equations (5.1), (5.2), and (5.3).

Example 5.2. Let I be the set N of all positive integers or the set Nk of ﬁrst k positive integers. Suppose (Ci)i∈I is a family of sets and A is the product of (Ci)i∈I in Set. If j ∈ J, n = nj, a1 ∈ A, . . . , an ∈ A, then ωj(a1, .., an) is a member of A deﬁned by

ai(i) if i ≤ n, ωj(a1, . . . , an)(i)= a1(i) otherwise, for any i ∈ I. Then A is an Ω-algebra satisfying (5.1), (5.2), and (5.3). For brevity, let us write Alg(n) for the category of all universal algebras with one n-ary operation. We describe objects of an essentially algebraic subcategory of Alg(n) having function spaces. They are precisely the ones of the type explained in the above example where I = Nn. Proposition 5.3. Suppose A is a subcategory of Alg(n) with canonical function spaces that contain the free object on Nn. Then for any A-object A there exist n Alg(n)-objects A1,...,An such that n-ary operation on Ai is n the i-th canonical projection Ai → Ai, and A is isomorphic to the product ′ A1 ×···× An in Alg(n) whose n-ary operation ω is given by ′ ω (a1, a2, . . . , an) = (a11, a22, . . . , ann), (5.4) where ai = (ai1, ai2, . . . , ain) for 1 ≤ i ≤ n. 228 V.L. Gompa

Proof. By Lemma (5.1), equations (5.1) and (5.3) hold in A. For any a ∈ A and i ∈ Nn, write Pia for the image of the product of A’s with {a} at the i-th place under the n-ary operation ω on A, i.e.,

Pia := {ω(z1,...,zn) : z1 ∈ A,...,zn ∈ A, zi = a}. (5.5)

Let Ai be the set of all Pia where a ∈ A, i.e.,

Ai := {Pia : a ∈ A}, (5.6) whose n-ary operation is the i-th canonical projection, and let B be the product of A1,...,An in Alg(n). We show that A is isomorphic to B. Deﬁne ϕ : B → A by

ϕ(P1a1,...,Pnan) := ω(a1, . . . , an) (5.7) for any a1 ∈ A, . . . , an ∈ A. To prove that ϕ is well deﬁned, note that

P1a1 ∩ . . . ∩ Pnan = {ω(a1, . . . , an)}. (5.8)

Indeed, clearly the set on the right is a subset of the set on the left, and if z is a member of the set on the left then there exist members zrs (1 ≤ r, s ≤ n) in A such that zrr = ar and z = ω(z1r, z2r, .., znr) for 1 ≤ r ≤ n. Now

ω(a1, . . . , an) = ω(z11,...,znn)

= ωj(ωj(z11,...,zn1),...,ωj(z1n,...,znn)) by (5.3) = ω(z,...,z)= z, by (5.1).

Note that from (5.7),

ϕ ◦ (P1 ×···× Pn)= ω, (5.9) and hence by (4.1),

ϕ ◦ (P1 ×···× Pn) ◦ dn,A = idA. (5.10) CARTESIANCLOSEDALGEBRAICCATEGORIES 229

In particular ϕ is onto. To show that ϕ is one-one, let us assume that ai ∈ A, ei ∈ A (1 ≤ i ≤ n), and ω(a1, . . . , an)= ω(e1,...,en). (5.11)

Fix i, 1 ≤ i ≤ n. Assume z1 ∈ A,...,zn ∈ A such that zi = ai. Write

ar, if s = i zrs = zs, if s 6= i and er, if s = i yrs = zs, if s 6= i for 1 ≤ r ≤ n, 1 ≤ s ≤ n. Then

ω(z1,...,zn) = ω(z11,...,znn)

= ω(ω(z11,...,zn1),...,ω(z1n,...,znn)) by (5.3)

= ω(ω(y11,...,yn1),...,ω(y1n,...,ynn)) by (5.11)

= ω(y11,...,ynn) by (5.3), which belongs to Piei. This shows that Piai is a subset of Piei. By interchanging the roles of ar’s and er’s, we see that

Piai = Piei.

This being true for any i, ϕ is one-one. By the deﬁnition of n-ary operation on B (see equation (5.4)) and by (5.3), it is easy to see that ϕ is a Ω-homomorphism.

Acknowledgments

The author wishes to acknowledge thanks to Dr. H. Lamar Bentley for his helpful suggestions. This work has been partially supported by Center for University Scholars.

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