Another characterization of congruence distributive varieties

TL;DR

This paper offers a new Maltsev-based characterization of congruence distributive varieties, establishing an equivalence with a specific congruence identity involving a finite number of factors.

Contribution

It introduces a novel Maltsev characterization of congruence distributive varieties through a specific congruence identity involving a finite number of factors.

Findings

Characterizes congruence distributive varieties via a congruence identity.

Establishes equivalence between Maltsev conditions and the identity.

Provides a new criterion for identifying such varieties.

Abstract

We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $\mathcal {V}$ is congruence distributive if and only if the congruence identity $\alpha \cap (\beta \circ \gamma \circ \beta ) \subseteq \alpha \beta \circ \gamma \circ \alpha \beta \circ \gamma \dots $ ($k$ factors) holds in $\mathcal {V}$, for some natural number $k$.

Equations

$$\alpha (\beta {\circ }_h\gamma )\subseteq \alpha \beta {\circ }_k\gamma$$

$$\alpha (\beta \circ \gamma )\subseteq \alpha \beta {\circ }_k\gamma ,\text{ for some k},$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \alpha \beta +\gamma$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \alpha \beta {\circ }_k\gamma .$$

$$\alpha (\beta {\circ }_h\gamma )\subseteq \alpha \beta {\circ }_p\gamma ,$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \alpha \beta {\circ }_r\alpha \gamma ,$$

$$(d_i(a,b,b,d),d_{i+2}(a,b,b,d))\in \alpha (\gamma \circ \alpha \beta \circ \gamma ),$$

$$d_i(a,b,b,d)\alpha d_i(a,b,b,a)=a=d_{i+2}(a,b,b,a)\alpha d_{i+2}(a,b,b,d),$$

$$d_{i+1}(a,b,c,d)\beta d_{i+1}(a,a,d,d)=d_{i+2}(a,a,d,d)\beta d_{i+2}(a,b,c,d),\text{ and}$$

$$d_{i+1}(a,b,c,d)\alpha d_{i+1}(a,b,c,a)=d_{i+2}(a,b,c,a)\alpha d_{i+2}(a,b,c,d),\text{ hence }$$

$$d_i(a,b,b,d)=d_{i+1}(a,b,b,d)\gamma d_{i+1}(a,b,c,d)\alpha \beta d_{i+2}(a,b,c,d)\gamma d_{i+2}(a,b,b,d),$$

$$(d_i(a,b,b,d),d_{i+2}(a,b,b,d))\in \alpha \gamma {\circ }_k\alpha \beta ,$$

$$d_{k-2}(a,b,b,d)=d_{k-1}(a,b,b,d)\alpha \beta d_{k-1}(a,a,d,d)=d_k(a,a,d,d)=d,$$

$$(a,d)\in \alpha \beta \circ \alpha \gamma \circ (\alpha \gamma {\circ }_k\alpha \beta {)}^{\frac{k-3}{2}}\circ \alpha \beta =\alpha \beta {\circ }_r\alpha \gamma ,$$

$$(d_1(a,b,c,d),d_3(a,b,b,d))\in \alpha \gamma \circ \alpha (\gamma \circ \alpha \beta \circ \gamma )=\alpha (\gamma \circ \alpha \beta \circ \gamma ),$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \gamma {\circ }_k\alpha \beta ,\text{ if k is even.}$$

$$(d_1(a,b,c,d),d_3(a,b,b,d))\in \alpha \beta {\circ }_k\alpha \gamma .$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \beta +\alpha \gamma$$

$$\alpha (\Delta \circ \gamma \circ \Delta )\subseteq \alpha \Delta {\circ }_{\kappa }\gamma ,$$

$$\alpha (\beta {\circ }_{2^{\ell }-1}\gamma )\subseteq \alpha \beta {\circ }_{2r^{\ell -1}}\gamma ,\text{ for every ℓ≥2. }$$

$$\alpha (\beta {\circ }_{2^{\ell +1}-1}\gamma )=\alpha (\Delta \circ \gamma \circ \Delta ){\subseteq }^{\text{eqref}nte}\alpha \Delta {\circ }_{2r}\gamma {\subseteq }^{\mathrm{ih}}(\alpha \beta {\circ }_{2r^{\ell -1}}\gamma ){\circ }_{2r}\gamma =\alpha \beta {\circ }_{2r^{\ell }}\gamma ,$$

$$\alpha (\beta {\circ }_{2^{\ell }-1}\gamma )\subseteq \alpha \beta {\circ }_{2^s+1}\alpha \gamma ,$$

$$\alpha (\beta \circ \gamma \circ \beta )\subseteq \alpha \beta {\circ }_{2^t+1}\alpha \gamma ,$$

$$\alpha (\beta {\circ }_{2^{\ell }-1}\gamma )\subseteq \alpha (\alpha \beta {\circ }_{2r^{\ell -1}}\gamma )=\alpha \beta \circ \alpha (\gamma {\circ }_{2r^{\ell -1}-1}\alpha \beta ),$$

$$\alpha (\gamma {\circ }_{2r^{\ell -1}-1}\alpha \beta )=\alpha (\gamma {\circ }_{2^q-1}\alpha \beta )\subseteq \alpha \gamma {\circ }_{2r^{q-1}}\alpha \beta$$

Full text

Work performed under the auspices of G.N.S.A.G.A. Work partially supported by PRIN 2012 “Logica, Modelli e Insiemi”. The author acknowledges the MIUR Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

We assume the reader is familiar with basic notions of lattice theory and of universal algebra. A small portion of [9] is sufficient as a prerequisite.

A lattice is distributive if and only if it satisfies the identity $\alpha(\beta+\gamma)\leq\alpha\beta+\gamma$. It follows that an algebra $\mathbf{A}$ is congruence distributive if and only if, for all congruences $\alpha$, $\beta$ and $\gamma$ of $\mathbf{A}$ and for every $h$, the inclusion $\alpha(\beta\circ_{h}\gamma)\subseteq\alpha\beta+\gamma$. holds. Here juxtaposition denotes intersection, $+$ is join in the congruence lattice and $\beta\circ_{h}\gamma$ is $\beta\circ\gamma\circ\beta\circ\gamma\dots$ with $h$ factors ($h-1$ occurrences of $\circ$).

Considering now a variety $\mathcal{V}$, it follows from standard arguments in the theory of Maltsev conditions that $\mathcal{V}$ is congruence distributive if and only if, for every $h$, there is some $k$ such that the congruence identity

[TABLE]

holds in $\mathcal{V}$. The naive expectation (of course, motivated by [4]) that the congruence identity

[TABLE]

is enough to imply congruence distributivity is false. Indeed, by [3, Theorem 9.11], a locally finite variety $\mathcal{V}$ satisfies (2) if and only if $\mathcal{V}$ omits types 1, 2, 5. More generally, with no finiteness assumption, Kearnes and Kiss [6, Theorem 8.14] proved that a variety $\mathcal{V}$ satisfies (2) if and only if $\mathcal{V}$ is join congruence semidistributive. Many other interesting equivalent conditions are presented in [3, 6].

In spite of the above results, we show that the next step is enough, namely, if we take $h=3$ in identity (1), we get a condition implying congruence distributivity. After a short elementary proof relying on [1, 4], in Remark 3 we sketch an alternative argument which relies only on [7]. Then, by working directly with the terms associated to the Maltsev condition arising from (1) for $h=3$, we show that this instance of (1) implies $\alpha(\beta\circ\gamma\circ\beta)\subseteq\alpha\beta\circ_{r}\alpha\gamma$, for some $r<\frac{k^{2}}{2}$.

[TABLE]

holds in every congruence lattice of algebras in $\mathcal{V}$.

For the nontrivial direction, assume that (3) holds in $\mathcal{V}$. By taking $\alpha\gamma$ in place of $\gamma$ in (3) we get $\alpha(\beta\circ\alpha\gamma\circ\beta)\subseteq\alpha\beta+\alpha\gamma$. Day [1] has showed that this identity implies congruence modularity within a variety. From (3) and congruence modularity we get $\alpha(\beta\circ\gamma\circ\beta)\subseteq\alpha(\alpha\beta+\gamma)=\alpha\beta+\alpha\gamma$ and, since trivially $\alpha(\beta\circ\gamma)\subseteq\alpha(\beta\circ\gamma\circ\beta)$, we obtain $\alpha(\beta\circ\gamma)\subseteq\alpha\beta+\alpha\gamma$. Within a variety this identity implies congruence distributivity by [4]. ∎

It is standard to express Theorem 1 in terms of a Maltsev condition.

1. (i)

$\mathcal{V}$* satisfies the congruence identity*

[TABLE] 2. (ii)

The identity (4) holds in $\mathbf{F}_{\mathcal{V}}(4)$, the free algebra in $\mathcal{V}$ generated by four elements $x,y,z,w$; actually, it is equivalent to assume that (4) holds in $\mathbf{F}_{\mathcal{V}}(4)$ in the special case when when $\alpha=Cg(x,w)$, $\beta=Cg((x,y),(z,w))$ and $\gamma=Cg(y,z)$. 3. (iii)

$\mathcal{V}$* has $4$-ary terms $d_{0},\dots,d_{k}$ such that the following equations are valid in $\mathcal{V}$:*

1. (a)

$x=d_{0}(x,y,z,w)$; 2. (b)

$d_{i}(x,x,w,w)=d_{i+1}(x,x,w,w)$, for $i$ even; 3. (c)

$d_{i}(x,y,z,x)=d_{i+1}(x,y,z,x)$, for $i$ even; 4. (d)

$d_{i}(x,y,y,w)=d_{i+1}(x,y,y,w)$, for $i$ odd, and 5. (e)

$d_{k}(x,y,z,w)=w$.

Clearly congruence distributivity implies the second statement in (ii), for some $k$; moreover identity (4) in (i) implies identity (3), hence congruence distributivity follows from Theorem 1. ∎

[TABLE]

hence also $\alpha(\beta\circ_{h}\gamma)\subseteq\alpha(\alpha\beta\circ_{p}\gamma)=\alpha\beta\circ\alpha(\gamma\circ_{p-1}\alpha\beta)$. Taking now $\gamma$ in place of $\beta$, $\alpha\beta$ in place of $\gamma$ and $p-1$ in place of $h$ in (5), we get $\alpha(\gamma\circ_{p-1}\alpha\beta)\subseteq\alpha\gamma\circ_{q}\alpha\beta$, for some $q$, thus $\alpha(\beta\circ_{h}\gamma)\subseteq\alpha\beta\circ_{q+1}\alpha\gamma$. In particular, $\alpha(\beta\circ_{h}\gamma)\subseteq\alpha\beta+\alpha\gamma$, for every $h$, hence we get congruence distributivity, by a remark at the beginning. Compare [8] for corresponding arguments. If one works out the details, one obtains that if $k\leq 2^{t}$, $t\geq 1$ and $\ell\geq 2$, then identity (4) implies $\alpha(\beta\circ_{2^{\ell}-1}\gamma)\subseteq\alpha\beta\circ_{2^{s}+1}\alpha\gamma$, with $s=(t-1)^{2}(\ell-1)+1$, a rather large number of factors on the right. We shall present explicit details in the Appendix.

We are now going to show that we can obtain a lighter bound on the right using different methods.

1. (f)

$x=d_{i}(x,y,y,x)$, for every $i\leq k$.

This follows immediately by induction from (a), (c) and (d). From the point of view of congruence identities, this corresponds to taking $\alpha\gamma$ in place of $\gamma$ in (3), as we did in the proof of Theorem 1. At the level of Maltsev conditions, this gives a proof that Clause (iii) in Corollary 2 implies congruence modularity, since the argument shows that the terms $d_{0},\dots,d_{k}$ obey Day’s conditions [1] for congruence modularity.

[TABLE]

where $r=\frac{k^{2}-4k+9}{2}$ for $k$ odd, and $r=\frac{k^{2}-3k+4}{2}$ for $k$ even.

[TABLE]

for every odd index $i<k-1$. Indeed,

[TABLE]

by (f) in the above remark. Moreover, still assuming $i$ odd,

[TABLE]

thus (6) follows. From (6) and (4) with $\gamma$ in place of $\beta$ and $\alpha\beta$ in place of $\gamma$, we get

[TABLE]

for every odd index $i$.

Arguing as above, $a\mathrel{\alpha\beta}d_{1}(a,b,c,d)\mathrel{\alpha\gamma}d_{1}(a,b,b,d)$. If $k$ is odd, then

[TABLE]

thus the elements $d_{1}(a,b,b,d),d_{3}(a,b,b,d),\dots,d_{k-2}(a,b,b,d)$ witness

[TABLE]

for $r=\frac{k^{2}-4k+9}{2}$. In the computation of $r$ we have used that, say, $(\alpha\gamma\circ_{k}\alpha\beta)\circ(\alpha\gamma\circ_{k}\alpha\beta)=\alpha\gamma\circ_{2k-1}\alpha\beta$, $(\alpha\gamma\circ_{k}\alpha\beta)^{3}=\alpha\gamma\circ_{3k-2}\alpha\beta$, etc., since $k$ is odd, hence there are adjacent occurrences of $\alpha\gamma$ which join into one. In the general case, $(\alpha\gamma\circ_{k}\alpha\beta)^{t}=\alpha\gamma\circ_{t(k-1)+1}\alpha\beta$, for $k$ odd. Finally, we have two adjacent occurrences of $\alpha\gamma$ at the second and third place in (7), too. From the above observations we get the value $\frac{k^{2}-4k+9}{2}$ of $r$.

On the other hand, if $k$ is even, then $d_{k-1}(a,b,b,d)=d_{k}(a,b,b,d)=d$. Moreover, since $d_{1}(a,b,c,d)\mathrel{\alpha\gamma}d_{1}(a,b,b,d)$, we have by (6)

[TABLE]

hence we can consider $d_{1}(a,b,c,d)$ in place of $d_{1}(a,b,b,d)$. By considering the converse of (4), we get

[TABLE]

Taking $\gamma$ in place of $\beta$ and $\alpha\beta$ in place of $\gamma$ in (9), then from (8) we get

[TABLE]

We can go on the same way, using alternatively (9) and (4) and considering the elements $d_{1}(a,b,c,d),d_{3}(a,b,b,d),d_{5}(a,b,b,d),\dots,d_{k-3}(a,b,b,d)$, getting $(a,d)\in(\alpha\beta\circ_{k}\alpha\gamma)\circ_{\frac{k-2}{2}}(\alpha\gamma\circ_{k}\alpha\beta)=\alpha\beta\circ_{r}\alpha\gamma$, for $r=\frac{k^{2}-3k+4}{2}$. ∎

We expect that the evaluation of $r$ in Theorem 5 can be further improved, but we have no guess as to what extent.

One can consider an identity intermediate between (2) and (1) by shifting the occurrence of $\alpha$ the other way, with respect to (4).

[TABLE]

We are not claiming that the above problem is difficult; in any case, it is not solved by the present note. As usual, a variety satisfies (10) if and only if there is some $k$ such that $\alpha(\beta\circ\gamma\circ\beta)\subseteq\beta\circ_{k}\alpha\gamma$ holds in $\mathcal{V}$. Let us also notice that the identity (10) implies congruence distributivity if and only if it implies congruence modularity. Indeed, if (10) implies congruence modularity, then we get distributivity arguing as in the last two sentences of the proof of Theorem 1.

The author considers that it is highly inappropriate, and strongly discourages, the use of indicators extracted from the list below (even in aggregate forms in combination with similar lists) in decisions about individuals (job opportunities, career progressions etc.), attributions of funds and selections or evaluations of research projects.

Appendix

In this appendix we justify the values reported in Remark 3.

Lemma 7**.**

The conditions in Corollary 2 are also equivalent to:

(iv) For every algebra $\mathbf{A}\in\mathcal{V}$, the following identity

[TABLE]

holds, for all congruences $\alpha$ and $\gamma$ on $\mathbf{A}$ and every tolerance $\Delta$ on $\mathbf{A}$ such that there exists an admissible relation $R$ on $\mathbf{A}$ for which $\Delta=R\circ R^{\smallsmile}$.

Proof.

The equivalence of (i) and (iv) is a special case of [7, Theorem 3 (i) $\Rightarrow$ (iii)]. For the reader’s convenience, we sketch a direct proof of (iii) $\Rightarrow$ (iv), while, of course, (iv) $\Rightarrow$ (i) is obvious.

So let us assume that we have terms as given by (iii) and that $\alpha$, $\Delta$ and $\gamma$ satisfy the assumptions in (iv). Suppose that $(a,d)\in\alpha(\Delta\circ\gamma\circ\Delta)$, thus $a\mathrel{\alpha}d$ and $a\mathrel{\Delta}b\mathrel{\gamma}c\mathrel{\Delta}d$, for certain $b,c\in A$. Moreover, by the assumption on $\Delta$, $a\mathrel{R}b^{\prime}\mathrel{R^{\smallsmile}}b$ and $c\mathrel{R}c^{\prime}\mathrel{R^{\smallsmile}}d$, for certain $b^{\prime},c^{\prime}\in A$. We claim that the elements $d_{i}(a,b,c,d)$, for $i=0,\dots,k$, witness that $(a,d)\in\alpha\Delta\circ_{\kappa}\gamma$. For example, let us check that $d_{i}(a,b,c,d)\mathrel{\Delta}d_{i+1}(a,b,c,d)$, for $i$ even. Indeed, $d_{i}(a,b,c,d)\mathrel{R}d_{i}(b^{\prime},b^{\prime},c^{\prime},c^{\prime})=d_{i+1}(b^{\prime},b^{\prime},c^{\prime},c^{\prime})\mathrel{R^{\smallsmile}}d_{i+1}(a,b,c,d)$, since, say, $b\mathrel{R}b^{\prime}$ and since, by assumption, $\Delta=R\circ R^{\smallsmile}$. All the rest is standard and simpler. We have proved that (i) - (iv) are equivalent, for every $k$. ∎

We now prove that (iv) implies $\alpha(\beta+\gamma)\leq\alpha\beta+\gamma$, an identity equivalent to distributivity. We shall actually show that if $k$ is even, say, $k=2r$, then (iv) implies

[TABLE]

Clearly, it is no loss of generality to assume that $k$ is even, since if the identity (11) holds for some odd $k$, then (11) holds for $k+1$, as well. Moreover, if $(a,b)\in\alpha(\beta+\gamma)$ in some algebra, then $(a,b)\in\alpha(\beta\circ_{2^{\ell}-1}\gamma)$, for some sufficiently large $\ell$ depending on $a$ and $b$. Hence, in order to show congruence distributivity, it is enough to prove the identity (12).

The proof of (12) is by induction on $\ell\geq 2$. The base case $\ell=2$ is the special case $\Delta=\beta$ of identity (11). Suppose that the identity (12) holds for some $\ell\geq 2$ and set $\Delta=\beta\circ_{2^{\ell}-1}\gamma$. By the inductive hypothesis, we have $\alpha\Delta\subseteq\alpha\beta\circ_{2r^{\ell-1}}\gamma$.

If $R=\beta\circ_{2^{\ell-1}}\gamma$, then $\Delta=R\circ R^{\smallsmile}$. Indeed, $\ell\geq 2$, thus $2^{\ell-1}$ is even, hence the last factor in the definition of $R$ is $\gamma$ and $\gamma$ is also the first factor of $R^{\smallsmile}$. Since $\gamma$ is a congruence, we have $\gamma\circ\gamma=\gamma$, namely, one factor absorbs in $R\circ R^{\smallsmile}$, thus $R\circ R^{\smallsmile}$ has $2^{\ell}-1$ factors, hence $\Delta=R\circ R^{\smallsmile}$. Thus we can apply (iv) and we have

[TABLE]

where the superscripts (11) and “ih” mean that we have applied, respectively, identity (11) and the inductive hypothesis and where in the last identity we have used again $\gamma\circ\gamma=\gamma$, noticing that $2r^{\ell-1}$ is even, hence the last factor in the expression $\alpha\beta\circ_{2r^{\ell-1}}\gamma$ is $\gamma$.

The induction step is thus complete, hence we have proved (12).

In the next corollary we state explicitly some informations which can be obtained from the above arguments.

Corollary 8**.**

If some variety $\mathcal{V}$ satisfies one of the equivalent conditions in Theorem 2 with $k\leq 2r$, then $\mathcal{V}$ satisfies the identity (12), for every $\ell\geq 2$.

If in addition $k\leq 2^{p}$, for some $p\geq 1$, then, for every $\ell\geq 2$, $\mathcal{V}$ satisfies

[TABLE]

where $s=(p-1)^{2}(\ell-1)+1$. In particular, taking $\ell=2$, we get that $\mathcal{V}$ satisfies

[TABLE]

for $t=(p-1)^{2}+1$.

Proof.

The first statement is given by the above proof of (12).

To prove the second statement, it is no loss of generality to assume that $k=2r$ and $k=2^{p}$. Notice that from (12) we get

[TABLE]

since $\alpha$ is supposed to be a congruence, in particular, transitive. From $k=2r$ and $k=2^{p}$, we get $r=2^{p-1}$, hence $2r^{\ell-1}-1=2^{q}-1$, for $q=(p-1)(\ell-1)+1$. Applying (12) with $\gamma$ in place of $\beta$, with $\alpha\beta$ in place of $\gamma$ and $q$ in place of $\ell$, we get

[TABLE]

and the conclusion follows from (13), since $2r^{q-1}=2(2^{p-1})^{(p-1)(\ell-1)}=2^{s}$. ∎