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Mal'cev variety

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Definition

A Mal’cev operation on a set XX is a ternary operation, a function

t:X×X×XX,(x,y,z)t(x,y,z), t:X\times X\times X\to X,\,\,\,(x,y,z)\mapsto t(x,y,z) ,

which satisfies the identities t(x,x,z)=zt(x,x,z)=z and t(x,z,z)=xt(x,z,z)=x. An important motivating example is the operation tt of a heap, for example the operation on a group defined by t(x,y,z)=xy 1zt(x, y, z) = x y^{-1} z.

An algebraic theory TT is a Mal’cev theory when TT contains a Mal’cev operation. An algebraic theory is Mal’cev iff one of the following equivalent statements is true:

  1. in the category of TT-algebras, every internal reflexive relation is a congruence;

  2. in the category of TT-algebras, the composite (as internal relations) of any two congruences as a congruence;

  3. in the category of TT-algebras, the composition of equivalence relations is commutative.

Statement (i) is one of the motivations to introduce the notion of Mal'cev category.

A Mal’cev variety is the category of TT-algebras for a Mal’cev theory TT, thought of as a variety of algebras.

Proofs of equivalence

If RX×YR \hookrightarrow X \times Y is a binary relation on sets, write R(x,y)R(x, y) to say that (x,y)R(x, y) \in R. If XX, YY are TT-algebras, then RR is an internal relation in TT-AlgAlg if the conditions R(x 1,y 1)R(x n,y n)R(x_1, y_1) \wedge \ldots \wedge R(x_n, y_n), and θ(x 1,,x n)=x\theta(x_1, \ldots, x_n) = x, θ(y 1,,y n)=y\theta(y_1, \ldots, y_n) = y for any nn-ary operation θ\theta of TT, jointly imply R(x,y)R(x, y).

The set-theoretic composite of two internal relations in TT-AlgAlg is also an internal relation, and the equality relation is always internal, so we may (and will) apply ordinary set-theoretic reasoning in our proofs below.

Proposition 1

If TT is a Mal’cev theory, then any internal reflexive relation in TT-AlgAlg is an internal equivalence relation.

Proof

If tt is a Mal’cev operation and RR is any internal reflexive relation on a TT-algebra XX, then RR is transitive because given R(x,y)R(y,z)R(x, y) \wedge R(y, z), we infer R(x,y)R(y,y)R(y,z)R(x, y) \wedge R(y, y) \wedge R(y, z), and this together with t(x,y,y)=xt(x, y, y) = x and t(y,y,z)=zt(y, y, z) = z gives R(x,z)R(x, z) since RR is internal. Also RR is symmetric, because if R(x,y)R(x, y), we infer R(x,x)R(x,y)R(y,y)R(x, x) \wedge R(x, y) \wedge R(y, y), which together with t(x,x,y)=yt(x, x, y) = y and t(x,y,y)=xt(x, y, y) = x gives R(y,x)R(y, x).

Proposition 2

If every internal reflexive relation is an internal equivalence relation, then the composite of any two internal equivalence relations is also an internal equivalence relation.

Proof

The hypothesis is that internal reflexive relations and internal equivalence relations coincide. But (internal) reflexive relations are clearly closed under composition: Δ=ΔΔRS\Delta = \Delta \circ \Delta \subseteq R \circ S.

Proposition 3

If internal equivalence relations are closed under composition, then composition of internal equivalence relations is commutative.

Proof

If RR and SS are equivalence relations and so is SRS \circ R, then

SR=(SR) op=R opS op=RS,S \circ R = (S \circ R)^{op} = R^{op} \circ S^{op} = R \circ S,

as desired.

Proposition 4

If composition of internal equivalence relations in TT-AlgAlg is commutative, then the theory TT has a Mal’cev operation tt.

Proof

According to the yoga of (Lawvere) algebraic theories, nn-ary operations are identified with elements of F(n)F(n), the free TT-algebra on nn generators (more precisely, the Lawvere theory is the category opposite to the category of finitely generated free TT-algebras). Thus we must exhibit a suitable element tt of F(3)F(3).

Let x,y,zx, y, z be the generators of F(3)F(3), and let a,ba, b be the generators of F(2)F(2). Let ϕ\phi be the unique algebra map F(3)F(2)F(3) \to F(2) taking xx and yy to aa and zz to bb, and let ψ\psi be the unique algebra map F(3)F(2)F(3) \to F(2) taking xx to aa and yy and zz to bb. An operation tF(3)t \in F(3) is Mal’cev precisely when

ϕ(t)=bψ(t)=a\phi(t) = b \qquad \psi(t) = a

Let RR be the equivalence relation on F(3)F(3) given by the kernel pair of ϕ\phi, and let SS be the kernel pair of ψ\psi. Then R(x,y)R(x, y) and S(y,z)S(y, z), so (SR)(x,z)(S \circ R)(x, z). Then, since composition of equivalence relations is assumed commutative, (RS)(x,z)(R \circ S)(x, z). This means there exists tt such that S(x,t)S(x, t) and R(t,z)R(t, z), or that ψ(x)=ψ(t)\psi(x) = \psi(t) and ϕ(t)=ϕ(z)\phi(t) = \phi(z). This completes the proof.

Examples

  • The theory of groups, where t(x,y,z)=xy 1zt(x, y, z) = x y^{-1} z, is Mal’cev.

  • The theory of Heyting algebras, where

    t(x,y,z)=((zy)x)((xy)z),t(x, y, z) = ((z \Rightarrow y) \Rightarrow x) \wedge ((x \Rightarrow y) \Rightarrow z),

    is Mal’cev.

  • If TT is Mal’cev, and if TTT \to T' is a morphism of algebraic theories, then TT' is Mal’cev. From this point of view, the theory of groups is Mal’cev because the theory of heaps is Mal’cev, and the theory of Heyting algebras is Mal’cev because the theory of cartesian closed meet-semilattices is Mal’cev.

See also Mal'cev category.

The lattice of congruences Equiv(X)Equiv(X)

Equiv(X)Equiv(X) is a modular lattice

In any finitely complete category, the intersection of two congruences (equivalence relations) on an object XX is a congruence, so that the set of equivalence relations Equiv(X)Equiv(X) is a meet-semilattice.

In a regular category such as a variety of algebras, where there is a sensible calculus of relations and relational composition, it is a simple matter to prove that if Equiv(X)Equiv(X) is closed under relational composition, then RSR \circ S is the join RSR \vee S in Equiv(X)Equiv(X). For, if R,SEquiv(X)R, S \in Equiv(X), then

R=RΔRS,S=ΔSRSR = R \circ \Delta \subseteq R \circ S, \qquad S = \Delta \circ S \subseteq R \circ S

while if R,STR, S \subseteq T in Equiv(X)Equiv(X), then

RSTTT.R \circ S \subseteq T \circ T \subseteq T.
Proposition

In a regular category, if Equiv(X)Equiv(X) is closed under relational composition (equivalently, if composition of equivalence relations is commutative), then Equiv(X)Equiv(X) is a modular lattice.

Proof

The (poset-enriched) category of relations in a regular category is an allegory, and hence satisfies Freyd’s modular law

R(ST)S((S opR)T)R \wedge (S \circ T) \subseteq S \circ ((S^{op} \circ R) \wedge T)

whenever T:XYT: X \to Y, S:YZS: Y \to Z, R:XZR: X \to Z are relations. As we have just seen, the hypothesis implies that joins in Equiv(X)Equiv(X) are given by composition (so Equiv(X)Equiv(X) is a lattice), and so for R,S,TEquiv(X)R, S, T \in Equiv(X) we have

R(ST)S((SR)T).R \wedge (S \vee T) \subseteq S \vee ((S \vee R) \wedge T).

Therefore, if SRS \subseteq R, we have both

R(ST)S(RT)R \wedge (S \vee T) \subseteq S \vee (R \wedge T)

and also S(RT)R(ST)S \vee (R \wedge T) \subseteq R \wedge (S \vee T). Thus SRS \subseteq R implies R(TS)=(RT)SR \wedge (T \vee S) = (R \wedge T) \vee S: the modular law is satisfied in Equiv(X)Equiv(X).

Corollary

If TT is a Mal’cev theory, then the lattice of congruences Equiv(X)Equiv(X) on any TT-algebra XX is a modular lattice.

Equiv(X)Equiv(X) is a Desarguesian lattice

A similar argument shows that congruence lattices for TT-algebras XX, for TT a Mal’cev theory, satisfy the following property (stronger than the modular property):

  • Desarguesian property?: if R i,S i,T iEquiv(X)R_i, S_i, T_i \in Equiv(X) for i=1,2i = 1, 2, then
    (R 1R 2)(S 1S 2))T 1T 2implies(R 1S 1)(R 2S 2)((R 1T 1)(R 2T 2))((S 1T 1)(S 2T 2))(R_1 \vee R_2) \wedge (S_1 \vee S_2)) \subseteq T_1 \vee T_2 \qquad implies \qquad (R_1 \vee S_1) \wedge (R_2 \vee S_2) \subseteq ((R_1 \vee T_1) \wedge (R_2 \vee T_2)) \vee ((S_1 \vee T_1) \wedge (S_2 \vee T_2))

Freyd-Scedrov’s Categories, Allegories (2.157, pp. 206-207) gives the following argument: given relations R 1,S 1,T 1:XYR_1, S_1, T_1: X \to Y, R 2,S 2,T 2:YZR_2, S_2, T_2: Y \to Z between sets, it is “easily verified” that

R 2R 1S 2S 2T 2T 1impliesS 1R 1 opS 2 opR 2(S 1T 1 opS 2 opT 2)(T 1R 1 opT 2 opR 2)R_2 R_1 \cap S_2 S_2 \subseteq T_2 T_1 \qquad implies \qquad S_1 R_{1}^{op} \cap S_{2}^{op} R_2 \subseteq (S_1 T_{1}^{op} \wedge S_{2}^{op} T_2)(T_1 R_{1}^{op} \cap T_{2}^{op}R_2)

Then, under the assumption that equivalence relations internal to TT-AlgAlg commute (so that the join of equivalence relations R,SR, S on XX is their relational composite RS=RSR S = R \circ S), the Desarguesian axiom follows immediately.

References

See the monograph Borceux-Bourn.

Spelling

The original is ‘Мальцев’; besides ‘Malʹcev’, this has also been transliterated ‘Malcev’ and ‘Maltsev’.

Revised on April 30, 2014 01:32:47 by Todd Trimble (67.81.95.215)