Algebra Universalis, 6 (1976) 225-228
Birkh~.user Verlag, Basel
Congruence modularity implies the Arguesian identity
RALPH FREESE AND BJARNI JONSSON
DEDICATED
T O R. P. D I L W O R T H
Abstract. It is shown that if V is any variety of algebras all of whose congruence lattices are modular,
then the congruence lattice of every algebra in ~ satisfies the Arguesian law.
1. Introduction
For any variety V of algebras, let Con (V) be the variety of lattices generated
by the congruence lattices Con (A) of the algebras A ~ ~. The purpose of this
paper is to show that if Con (~) is modular, then it is Arguesian. It was first
proved by J. B. Nation in [4] that certain varieties of lattices are not equal to
Con (~) for any variety V" of algebras, and since then his list of excluded varieties
has been considerably extended by various investigators. The variety of all
modular lattices was added to that list by the first author in a result announced in
[1], stating that if Con (V) is modular, then it satisfies a certain identity related to,
but apparently weaker than, the Arguesian law. His argument was modified by
the second author to yield the full Arguesian law. Actually we prove the dual of
the Arguesian law, but as was shown in [2], the two properties are equivalent.
2. The basic construction
The central idea of our argument is to e m b e d Con (A) in two different ways in
the congruence lattice of another algebra B e V. This will allow us to imitate the
This work was supported in part by NSF grants Nos. 73-08589 A02 and GP-29129 A-3.
Presented by G. Gr~itzer. Received November 20, 1975. Accepted for publication in final form
January 12, 1976.
225
226
RALPH FREESE AND BJARNI J6NSSON
ALGEBRA UNIV.
classical proof of Desargues' Law for projective planes embeddable in higher
dimensional projective spaces.
Actually the algebra B depends on a fixed congruence relation a e Con (A)
and is completely determined by a. In fact, we let
B = {(ao, al) e A x A : aoaal}.
Thus B is a subalgebra of A x A. Let "Oo and ri~ be the kernels of the projections
(Co, c~)--~ Co and (Co, c~)--~ c~ of B onto A, and let fo and fl be the induced
isomorphisms of Con (A) onto the intervals 1/rio and 1/ri! in Con (B). Finally, let
0 ~ O' be the involutionary automorphism of Con (B) induced by the automorphism (ao, al} ~ (al, ao) of B. More explicitly, for (ao, al}, (bo, bl)~ B, A ~ Con (A)
and 0 ~ Con (B),
(ao, aOrii(bo, bl)
iff
(ao, al)f,(h)(bo, bl)
al = b.
iff
a,hb,,
(ao, at)O'(bo, bl) iff {al, ao)O(b,, bo).
Observe that
(f~(A))' = f,_,(h),
fo(h)=fl(h)
whenever
a___h.
Letting ~" = fo(a)= fl(Ot), we thus have
0' = 0
whenever
~'_~ 0.
We also notice that
-n'= rio+rib
0 + 7r = O' + rio
whenever
rio--G0.
In fact, if (ao, a~)~-(bo, bl), then ao, ax, bo, bl are in the same a-class, so that
(ao, bl)~ B .and (ao, al)rio(ao, bl)ri~(bo, bl). This proves that , r ~ rio+ ri1, and the
opposite inclusion is obvious. Since ~r__q'0o+rh__q 0 + r i b we have that 0 + T r =
0 + ri|. Hence, since ' preserves all congruences which contain zr, 0 + 7r = (0 + ,r)' =
(0 + ri|)'= 0'+ rio.
Vol 6, 1976
Congruence modularity implies the Arguesian identity
227
3. Central and axial perspectivity
T h e following l e m m a is a slight modification of L e m m a 2 in Gr~itzer, J6nsson
and L a k s e r [2]. The p r o o f of that l e m m a carries over essentially u n c h a n g e d , and is
therefore omitted.
L E M M A A. Suppose L is a modular lattice and a = ( a o , al, a2) and b =
(bo, bb b2) are centrally perspective triangles in L, whose center of perspectivity p
satisfies the conditions
p+at=p+bi=ai+b~
for
Let u = a o + a l + a 2 + b o + b ~ + b 2 .
p + q = p + r = q + r,
i=0,1.
If there exist q, r ~ L such that
uq = pao,
ur = pbo,
then a and b are axially perspective.
4. The main theorem
W e now apply the construction from Section 2 to obtain a situation to which
L e m m a A can be applied.
L E M M A B. Suppose 3c is a variety of algebras, and let X be the class of all
lattices L such that L is embeddable in the dual of C o n (A) for some A ~ V. For
any L e Sg and p, s, t, u ~ L, if p + s = p + t = s + t<-u, then L has an extension
L' ~ Y{ such that for some q, r ~ L',
p+q=p+r=q+r,
qu =ps,
ru=pt.
Proof. Choose a dual e m b e d d i n g g : L - - ~ Con (A) with A 6 ~ , and let a, or, T
and u be the images of p, s, t and u u n d e r g. Construct B, r/o, "ql, 7r, fo and fl as in
Section 2, and let cri=fi(cr), l"i=fi(~ -) and v i = f i ( v ) . The dual e m b e d d i n g
f o g : L "-~ 1/r/o can be e x t e n d e d to a dual isomorphism h of an extension L' of L
onto C o n (B), and we take q and r to be the m e m b e r s of L' such that h ( q ) = o"1
and h ( r ) = 1"1. Since or1 = o,~ and r/o~ O'o, we have or1+ "qo = ~ o + zr, and using the
fact that -Ooc_ Uo~ O-o (because 1 -> u >- s), we infer that o't + Uo= O'o+ zr. A p p l y i n g
h -1 to b o t h sides of this e q u a t i o n we find that qu = sp. Similarly, ru = tp. Finally,
applying the dual e m b e d d i n g f~ g: L ~ 1/'ql to the equations p + s = p + t = s + t,
228
RALPH FREESE AND BJARNI J(~NSSON
we obtain 7r i"1o'1 = 7r f3 71 = ~rl t3 ~-1, and taking images under h -1 we conclude
that p + q = p + r = q + r .
T H E O R E M . For any variety ~ of algebras, if Con (~) is modular, then it is
Arguesian.
Proof. By J6nsson [3], the Arguesian identity is selfdual, and it is therefore
sufficient to show that, for 5~ as in L e m m a B, every lattice L in ~ is Arguesian.
By L e m m a 1 of Gr~itzer, J6nsson and Lakser [2], a modular lattice L is Arguesian
if any two triangles a and b in L that satisfy the conditions of L e m m a A are
axially perspective. T o prove that this property holds, we apply L e m m a B with
s = ao and t = bo, and then apply L e m m a A with L replaced by L'.
It is interesting to note that the algebra B in our construction is simply the
congruence relation a itself, viewed as a subalgebra of A • A. In fact, our proof
yields the following stronger local version of the main theorem.
C O R O L L A R Y 1. For any algebra A, if Con (~) is modular for every a 9
Con (A), then Con (A) is Arguesian.
C O R O L L A R Y 2. If ~ is an equational class such that Con (A) is modular for
every finite A 9 ~, then Con (A) is Arguesian for every finite A 9 ~.
REFERENCES
[1] RALPH FREESE, Congruence modularity, Preliminary Report, Notices Amer. Math. Soc. 22 (1975),
p. A-301.
[2] G. GRATZER, B. JONSSON and H. LAKSER, The amalgamation property in classes of modular
lattices, Pacific J. of Math. 45 (1973), 507-524.
[3] B. JONSSON, The class of Arguesian lattices is selfdual, Algebra Universalis 2 (1973), p. 396.
[4] J. B. NATION, Varieties whose congruences satisfy certain lattice identities, Algebra Universalis 4
(1974), 78-88.
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