Theory and Applications of Categories, Vol. 11, No. 9, 2003, pp. 212–214.

CHARACTERIZATION OF POINTED VARIETIES OF UNIVERSAL ALGEBRAS WITH NORMAL PROJECTIONS

ZURAB JANELIDZE

ABSTRACT. We characterize pointed varieties of universal algebras in which (A × B)/A ≈ B, i.e. all product projections are normal epimorphisms.

1. Definition. We will say that a pointed category C has normal projections if every product projection A × B → B in C is a normal epimorphism.

Equivalently, for any two objects A and B in such a category C, forming the product A×B and then factoring it by A ≈ A×0 results in B. In particular, every Jónsson-Tarski variety of universal algebras [3] (considered as a category) has this property; the same is true for the pointed subtractive varieties in the sense of Ursini [4].

The purpose of this paper is to characterize pointed varieties with normal projections (Theorem 3 below).

Before stating the theorem, we make a simple reformulation of Definition 1.

2. Proposition. Let C be a pointed variety. The following conditions are equivalent:

(a) C has normal projections;

(b) there exists a natural number n, such that for all A and B in C, and for all a ∈ A and b ∈ B, ((a, b), (0, b)) ∈ R^{n}, where R is the reflexive homomorphic relation on A × B generated by the set {((a′, 0), (a′′, 0))| a′ = 0 or a′′ = 0};

(c) let F [x] be the free algebra in C generated by x; there exists a natural number n such that ((x, x), (0, x)) ∈ Q^{n}, where Q is the reflexive homomorphic relation on F [x] × F [x] generated by the set {((x, 0), (0, 0)), ((0, 0), (x, 0))}.

Moreover, the number n in (b) and in (c) can be supposed to be the same.

3. Theorem. A pointed variety C has normal projections if and only if the corresponding theory contains

• unary terms t_{1}, ..., t_{m} and u_{1}, ..., u_{m};

• (m + 2)-ary terms v_{1}, ..., v_{n}; and the following identities hold in C:

Received by the editors 2003-01-08 and, in revised form, 2003-05-14.

Transmitted by Walter Tholen. Published on 2003-05-20.

2000 Mathematics Subject Classification: 18A20, 18A30, 08B05, 08B25.

\begin{matrix}c \\ ◯ \end{matrix} Zurab Janelidze, 2003. Permission to copy for private use granted.

• x = v_{1}(t_{1}(x), ..., t_{m}(x), x, 0);

• v_{i+1}(t_{1}(x), ..., t_{m}(x), x, 0) = v_{i}(t_{1}(x), ..., t_{m}(x), 0, x) for each i ∈ {1, ..., n − 1};

• 0 = v_{n}(t_{1}(x), ..., t_{m}(x), 0, x);

• x = v_{i}(u_{1}(x), ..., u_{m}(x), 0, 0) for each i ∈ {1, ..., n}.

Moreover, for each n this characterizes the pointed varieties of universal algebras which satisfy 2(b) (for the same n).

Proof. According to Proposition 2, we have to characterize pointed varieties for which ((x, x), (0, x)) ∈ Q^{n}, i.e. for which there exist s_{1}, ..., s_{n+1} ∈ F [x] such that s_{1} = x, s_{n+1} = 0, and for each i ∈ {1, ..., n} the pair ((s_{i}, x), (s_{i+1}, x)) is in Q. On the other hand, ((s_{i}, x), (s_{i+1}, x)) ∈ Q if and only if for some (m+2)-ary term v_{i} and unary terms t_{1}, ..., t_{m}, u_{1}, ..., u_{m} ∈ F [x], one has the equalities (s_{i}, x) = v_{i}((t_{1}, u_{1}), ..., (t_{m}, u_{m}), (x, 0), (0, 0)) and (s_{i+1}, x) = v_{i}((t_{1}, u_{1}), ..., (t_{m}, u_{m}), (0, 0), (x, 0)). Moreover, we can assume that the t’s, u’s and the number m are the same for each i ∈ {1, ..., n}. Writing the equalities above separately for the components of pairs, we obtain s_{i} = v_{i}(t_{1}, ..., t_{m}, x, 0), s_{i+1} = v_{i}(t_{1}, ..., t_{m}, 0, x), x = v_{i}(u_{1}, ..., u_{m}, 0, 0). Since s’s are expressed by v’s, we may omit them; after this the identities become exactly as in the formulation of the theorem.

4. Example. Let C be a pointed variety with normal projections, for which we could take n = 1 in Theorem 3. Then, the theory corresponding to C has unary terms t_{1}, ..., t_{m}, u_{1}, ..., u_{m} and an (m + 2)-ary term v, which satisfy the identities

x = v(t_{1}(x), ..., t_{m}(x), x, 0), x = v(u_{1}(x), ..., u_{m}(x), 0, 0), 0 = v(t_{1}(x), ..., t_{m}(x), 0, x).

When the unary terms are either x or 0, an easy argument shows that we could rewrite these identities as

x = w(x, 0, x, x, 0), x = w(x, x, 0, 0, 0), 0 = w(x, 0, x, 0, x).

If the term w = w(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) depends only on the first variable x_{1} and the last variable x_{5}, then we can write w(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = x_{1} + x_{5} and our identities become

x = x + 0, 0 = x + x;

in this case the variety C becomes nothing but a pointed subtractive variety in the sense of Ursini [4]. On the other hand, w(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = x_{2} + x_{4} would give

x = 0 + x, x = x + 0

which defines a Jónsson-Tarski variety [3].

5. Remark. It is known that every semi-abelian category has normal projections (see Condition SA*3a in [2]). More generally, every pointed category in which every pair of canonical morphisms (1_{A}, 0) : A → A × B, (0, 1_{B}) : B → A × B is jointly epimorphic has this property. In particular, this is the case for the unital categories in the sense of Bourn [1].

References

[1] D. Bourn, Mal’cev categories and fibration of pointed objects, Applied Categorical Structures 4, 1996, 302-327.

[2] G. Janelidze, L. Márki and W. Tholen, Semiabelian categories, Journal of Pure and Applied Algebra 168, 2002, 367-386.

[3] B. Jónsson and A. Tarski, Direct decompositions of finite algebraic systems, Notre Dame Mathematical Lectures, 1947, Notre Dame, Indiana.

[4] A. Ursini, On subtractive varieties, I, Algebra Universalis 31, 1994, 204-222.

The Faculty of Mechanics and Mathematics, Ivane Javakhishvili State University of Tbilisi, 1 Chavchavadze Ave., 380028 Tbilisi, Georgia

Email: zurab@gol.ge

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