A complete logic forn-permutable congruence lattices

Algebra Universalis, 13 (1981) 206-224 0002-5240/81/002206-19501.50 + 0.20/0 �9 1981 Birkh~iuser Verlag, Basel

A complete logic for n-permutable congruence lattices

GEORGE HUTCHINSON

Abstract. A family of logical systems, which may be regarded as extending equational logic, is studied. The equations f= g of equational logic are generalized to congruence equivalence formulas f=- g (rood x), where f and g are terms interpreted as elements of an algebra V of some specified type, and term x is interpreted as a member of an n-permutable lattice of congruences for V. Formal concepts of proof and derivability from systems of hypotheses are developed. These proofs, like those of equational logic, require only finite algebraic processes, without manipulation of logical quantifiers or connectives. The logical systems are shown to be correct and complete: a well-formed statement is derivable from a system of hypotheses if and only if it is valid in all models of these hypotheses.

w Introduction

W e wish to e x t e n d the logic of a lgebra i c equa t ions f = g (see [ 1: Thm. 10, p.

4 4 l ] ; a r ecen t expos i t ion is in [3: pp, 170-171]) . Both e q u a t i o n a l logic and our

ex t ens ion to congruences work f rom the a s sumpt ion that an a lgebra i c t ype r has

been specif ied, consis t ing of a rb i t ra r i ly m a n y ope ra t ions , each having a given finite

ar i ty . T h e w e l l - f o r m e d fo rmulas of equa t i ona l logic a re fo rma l equa t ions :

f(al, a2 . . . . . ak ) = g ( a l , a2 . . . . . ak)

for r - p o l y n o m i a l s f and g in s o m e c o n v e n i e n t d e n u m e r a b l e set of va r i ab les . In

o u r logic of congruences , the t e rms f and g of the f o r m a l c o n g r u e n c e f -=

g (mod x) a re again r - p o l y n o m i a l s for s o m e given r. H o w e v e r , the t e rm x =

x(e t , e2 . . . . . e i) is i n t e n d e d to r e p r e s e n t a r - a l g e b r a congruence . Since the set

Con (V) of congruences of a r - a l g e b r a V is a b o u n d e d la t t i ce u n d e r inclus ion, the

c o n g r u e n c e t e rm x is a b o u n d e d la t t ice po lynomia l . Specif ical ly, x is g e n e r a t e d

f rom a d e n u m e r a b l e set of va r i ab les d i s jo in t f rom the va r i ab le s used for r -

po lynomia l s , us ing mee t , jo in and nul la ry O and I o p e r a t i o n s . T h e w e l l - f o r m e d

f o r m u l a s of o u r logic of congruences are t r ip les of fo rm (f, g, x); this is on ly a

n o t a t i o n a l change f rom the use of fo rmal cong ruences f = g (mod x) o r fo rma l pa i r

m e m b e r s h i p (f, g) ~ x.

Presented by G. Gr~itzer. Received May 9, 1978. Accepted for publication in final form April 1 l, 1980.

206

Vol. 13, 1981 Logic for n-permutable congruences 207

In order to make our logic of congruences complete, we have found it necessary to impose an n-permutabil i ty restriction. Recall that for any congruences x and y of a z-algebra V, the join of congruences is the same as the join of parti t ions (see [3: I_emma 2, p. 50]). That is, we have the join formula:

x v y = U 2 ( x y x ". ").1,

where the m-fold alternating composition of the relations x and y on V is denoted as shown:

( x y x " ' ) m = z ~ z 2 " ' z , , . where z~=x f o r i o d d and

z i = y f o r i e v e n , i = 1 , 2 . . . . . m.

The formula expresses x v y as an infinite union of an ascending chain of relations on V, and so this formula is unmanageable by our finite logical processes. Accordingly, we assume that for some fixed n >--2, all pairs of congruences represented in our formulas satisfy the n-permutabil i ty formula:

x v y = ( x y x �9 �9 .),~.

The n-permutabi l i ty condition sets a known, finite bound upon the evaluation of congruence joins by relation composition, permitting a complete logic with proofs obtained from finite sequences of triples.

The models for an equational logic are certain z-algebras. For our logic of congruences, a model consists of a z-algebra V and an n-permutab le sublattice of Con (V) containing O and L Here , we don ' t insist that C o n ( V ) itself be n- permutable. For example, we can model n -permutab le sublattices of infinite lattices of partitions as were considered by B. J6nsson in [5], although the lattice of all partitions of an infinite set is not n-permutable for any n. However , many readily apparent applications of our logic of congruences are to varieties ~V of ~--algebras which have the proper ty that for some fixed n, Con (V) is n- permutable for all V in ~V. Recall the well-known theorem of A. I. Mal 'cev ([6], or see [3: Thm. 4, p. 172]) that all members of a variety "V of z-algebras have 2-permutable congruence lattices if and only if there is a z-polynomial p ( x , y, z) such that p ( x , y, y) = x and p(x, x, y) = y are identities for ~ . This was generalized by R. Wille [8: Satz 6.17, p. 78], who gave for each n ---2 a Mal 'cev condition for n-permutabi l i ty of congruence lattices for all algebras of a variety ~ . Many historically important algebraic varieties have a binary operat ion inducing a group

208 GEORGE HUTCHINSON ALGEBRA UNIV.

structure with respect to some unary inverse operation. Consider multiplication for any variety of groups, addition for any variety of rings, modules or associative algebras, the symmetric difference operation for Boolean algebras, and so on. Varieties with such a group operation have 2-permutable congruences by Mal'cev's theorem (p(x, y, z)= xy-~z), so a complete logic of congruences is avail- able for such theories. For varieties with a group operation, it is more convenient to work with homomorphism kernels rather than with congruences. So, a group and its lattice of normal subgroups, a ring and its lattice of two-sided ideals, or a module and its lattice of submodules may each be taken as a model for a suitable complete logic of congruences. The logic can be somewhat simplified by replacing formal congruences f ~ g (mod x) by homomorphism kernel memberships h ~ x in this case, because the congruence f-= g (mod x) is equivalent to membership of fg-~ (or f - g ) in the kernel corresponding to x for such varieties. Due to space limitations, we omit the discussion of this adaptation.

It is helpful to contrast our methods with the basic completeness theorem for equational logic. Recall that the equation:

f(a, , a ~ , . " , ak)= g(a,, a 2 , " ' , a~)

of equational logic is regarded as an algebraic identity: we implicitly associate a universal quantifier with each variable ai. The models of an equational logic are ~--algebras of a variety T', where ~g is specified by a given set v of ~--algebra identities. A proof in equational logic is a sequence of formal equations, such that each is an axiom or a consequence of previous equations by some rule of inference. The axioms are either reflexive equations f = f or substitution instances of identities in v. The rules of inference correspond to symmetry and transitivity for equations, plus the congruence property rule that any r -opera t ion applied to pairwise equal arguments has equal results. An equation is derivable if it is the last term of a proof. The completeness theorem states that an equation is derivable in a logic for T" if and only if it is an identity for every T-algebra in ~ .

Our logic of congruence formulas allows several different kinds of hypotheses. First, we can consider quasivarieties of T-algebras or bounded lattices, that is, classes which are axiomatizable by sets of formulas, each formula being either an identity f = g or the universal closure of an implicational open formula:

(fl = g, &f2 = g2 & ' " & f.~-, = g.~-,) f f fm= g.~.

where f~ and g~ are polynomials for each i -<__ m. Our models consist of a r-algebra V and an n-permutable sublattice L of Con (V). We may assume that (1) V belongs to a specified quasivariety of z-algebras, and (2) L belongs to a specified


quasivariety of bounded lattices. We may describe L by generators and relations, assuming that (3) certain named elements of L satisfy a specified set of lattice equations. Furthermore, we may assume (4) that each member of an arbitrary set A of triples (f, g, x) is an axiom, that is, postulate an arbitrary fixed set of congruences (within certain limits). Finally, we must assume that (5) for some specified n _-__ 2, L is n-permutable. Given a system of hypotheses of these five types, proofs are now sequences of triples (f, g, x> obtained from appropriate axioms and rules of inference. Again, derivable triples can be defined as the final terms of proofs, but there is a complication here. There is a rule of inference for n-permutability, which allows us to simultaneously infer, from one triple (/, g, x v y), a set of n triples of the form:

(L dt, x),(d,, dz, y) ,(d2, d3, x) . . . . . (dn_l, g ,z ) ,

where

z is x for n odd and y for n even.

Here dr, d2 . . . . . d.,_, are distinct z-algebra variables which have not appeared anywhere in the proof sequence to be extended by the above n-permutability rule. Essentially, the variables di are "rule C" variables of logic (see [7: pp. 123- 133]), with implicit existential quantifiers. Thus, the triples (f, g, x) in congruence logic proofs correspond to V3-sentences in the predicate calculus, but the final "derivable" triple must not contain any of these rule C variables d~, and so the derivable triple corresponds to a universal sentence.

Clearly, z-algebra equations f = g can be represented by the congruence (f,g, O> in our logic. However, a special device must be used to represent inclusions and equations between congruences. Unary function letters a and b are introduced, such that a(x) and b(x) are r-algebra expressions whenever x is a bounded lattice polynomial. These expressions act like z-algebra variables which satisfy the congruence a(x)=-b(x)(mod x), but can not be directly restricted in any other way. That is, (a(x), b(x)> represents a "generic pair" of r-algebra elements belonging to x. So, derivation of the triple (a(x), b(x), y> corresponds to a proof that any pair of elements of x must also belong to y, that is, to x c y. So, the equation x = y corresponds to the two triples (a(x), b(x), y> and (a(y), b(y), x>.

In w we develop a "s tandard" lattice representation method. To suggest its nature, we consider an example: Suppose a lattice L is given, and we want to know whether there exists any group G such that L can be embedded in the lattice of normal subgroups of G. Using a standard construction, we obtain a particular lattice homomorphism hL:L ~ Con (F), where F is a free group with


"sufficiently many" generators and )q_ is constructed from L by a specific procedure. Then )~c is one-one if and only if there exists G such that L is embeddable in the lattice of normal subgroups of G. In general, given a lattice L, a quasivariety ~ of ;r-algebras, and some n _-__ 2, a standard representat ion deter- mines a particular algebra V of "V and lattice homomorphism ) t :L- -~ Con (V) such that A[L] is an n-permutable sublattice of Con (V). The theorem concluding w shows that A is one-one if and only if for some U in "V there exists a lattice embedding t~ :L--*Con (U) such that ~ [L] is n-permutable. So, existence of certain lattice representations in congruence lattices can be reduced to examina- tion of a particular construction. The standard representation technique and the theorem of w can also accommodate optional constraints in the form of a specified set of triples A, Standard representations are a sort of algebraic adaptation of the set-theoretic constructions of B. J6nsson [5]. They are related also to the R-module sublattice constructions of [4].

In w we develop the logic of congruences and prove it correct and complete. The verification of correctness, that derivable formulas correspond to true statements in any model, is by induction on lengths of proofs as usual. To prove completeness, that any formula valid in all models is derivable, a model is constructed such that no non-derivable formula holds for this particular model. This construction, using the derivable triples and the standard representation method of w is similar to the construction of "F-free algebras to prove the completeness of an equational logic for 7/'.

w Standard representations in congruence lattices.

Suppose )t :L ---> Con (V) is a meet homomorphism preserving I for some (O, I) lattice L and r-algebra V. Then the set S of triples (u, v , x ) in V 2 x L (V2= V x V) such that u = - v ( m o d A ( x ) ) has the following properties, for all u , v , w in V a n d x , y in L:

(1 ) (u, v, x )~ S and (u, v, y)~ S ~ (u, v, xA y)~ S. (2) x c y in L and ( u , v , x ) E S ~ ( u , v , y ) ~ S . (3) (u, v, I )~S . (4) (u, u , x ) 6 S . (5) (u ,v ,x)eS~(v, u,x)~S. (6) (u, v, x)~ S and (v, w, x)~ S ~ (u, w, x)~ S. (7) If p(a~, a2 . . . . . ak) is a r -operat ion and (u~, v~ ,x )~S .for i = 1, 2 . . . . . k,

then:

@(ul, u2 . . . . . Uk), p(V,, V2 . . . . . V~), X)~ S.


An "MHI-sys tem" S on V2x L is a subset of Vax L satisfying (1)-(7) above. Clearly, the family ~ of all MHI-systems on V2xL is a closure system [3: pp. 23-24], closed for arbitrary intersections and containing V 2 x L itself. Let 5r denote the MHI-system generated by an arbitrary subset X of V 2 x L. Since S is in ~ i f f S e ( X ) c S for all finite subsets X of S, o~ is an algebraic closure system. For subsets A of V 2 and B of L, define "polars" for the Galois connection associated with each S in ,~ as usual:

A * = { x ~ L : ( u , v , x ) ~ S fo ra l l (u ,v)~A}

and

B~f={(u,v)eV2:(u,v,x)eS fo ra l l xeB} .

(See [2: pp. 122-125].) By computing, we see that each polar A* is a dual ideal in L, and each polar B t is in Con (V). Also, )t s : L ~ Con (V) with )ts(x) ={x}t is a meet homomorphism preserving /. Since the construction of ,k s from S is reciprocal to the construction of an MHI-system from a meet homomorphism preserving I given above, there is a one-one correspondence between MHI- systems on V2x L and meet homomorphisms L--->Con (V) preserving L

We now introduce the standard lattice representation method.

D E F I N I T I O N 1. Suppose L is a (O, 1) lattice, n ---2, and T" is a quasivariety of ~--algebras axiomatized by a set X of identities and implicational formulas. Let A be any subset of T'{B}2x L, where T'{B} is the Y-free algebra on some set B.

We intend to construct an MHI-system S c T'{A}2• L for a set A ~ B, such that S induces a (0, I) lattice homomorphism with n-permutable image. Now, S induces a meet homorphism preserving I automatically, and we can force preser- vation of 0 by taking the quotient of 7/'{A} by the congruence {O}t. Say that ( f ,g,x,y) in "V{A}2xL 2 is a "defect" for S if ( / ,g) is in { x v y } t but not in ({x}t{y}t{x}t"- ")~. So, S has no defects if and only if

{x v y } t = ( { x } t { y } t { x } t �9 �9 - ) .

holds for all x, y in L, which is precisely the condition needed so that S induces a join homomorphism with n-permutable image. Finally, we want 7;{A}/{O}t to belong to 7/'; this is always true if "F is a variety, But suppose X contains the formula ~ below:

(U l = 1) 1 t~ U2 = ' / ) 2 I~ " ' " t ~ Um--I = "/3ra-l) ~ Urn = ?")m,

212 G E O R G E HI.YFCHINSON A L G E B R A UNIV.

and -q is an assignment of values in T'{A} to the variables of w, which can be regarded as a q--homomorphism for all T-polynomials on these variables. Then (~, r/) is a "faul t" for S if (r/(u~), aq(v/), O) is in S for i = 1 , 2 . . . . . m - l , but (r/(u~), "O(v,,), O) is not in S. It is easily checked that "F{A}/{O}t is in ~ if and only if S has no faults. We construct the standard system S by a transfinite induction, eliminating defects and faults one step at a time. These steps are "minimal" in an appropriate sense, so that the standard representation has the required characteri- zation property.

Let p be an infinite cardinal number, not smaller than the cardinality of r, of L or of B. The set of generators A is provided with certain elements a~ and b~ for each x in L, corresponding to the "generic pairs" in the logical theory, and also generators a~, i <_- n - 1 and v an ordinal number less than 19, which are used to correct defects. That is, A = Ap as given below:

Ao = B U{ax: x ~ L} U{b.: x e L}.

A , = A o U { a ~ , , : t x < u , l < = i < = n - 1 } for O < v < - o .

(All elements ax, bx and a~ are assumed distinct and not in B.) Let T denote the set "V{A}2 • L 2. Let U denote the set of pairs (~, 7/) such that

is an implicational formula in v and -q is an assignment of values in 7/'{A} to variables occurring in ~ (and by ~--homomorphism extension, to r-polynomial terms of ~) . Then T and U both have cardinality at most O, and so we can well-order T and U so that each element of T has fewer than p predecessors in the well-ordering for T, and similarly for U.

We say that f in T'{A} has "suppor t" on X c A if f is in T'{X}; clearly every f in ~V{A} has finite support, and so has support on T'{A~} for some u < p. For u <= p, define Y~ = 'F {A}2• by transfinite induction up to p, given by:

(1) Yo=A U{(ax, b,, x ) : x ~ L } .

(2) If v ~ p is a limit ordinal, let Y~ = U~<v Y~. (3) For successor ordinals, Y~+I = Yv L_J T~ t_l U~ if v < o , where T~ and U~ are

defined next. If (f, g, x, y) is the smallest element in the well-ordering for T which is a defect for S~ and such that f and g have support on A~, then:

T~ = {(f, al~, x), (a,~, a2~, y), (a2~, a3~, x) . . . . . (a._i.~, g, z)},

where z is x for n odd, y for n even.

If there is no such (f, g, x, y), then T~ is empty. If ( w .q) is the smallest element in the well-ordering for U which is a fault for ~(Y~) and such that 7l(c) has support


on A, for all variables c of _~, then U~ = {(-q(um), "q(vm), 0)}, where u~ = vm is the conclusion of the implicational formula _~. If there is no such (_~, ~), then U~ is empty.

This completes the standard representation construction; the limit MHI- system S = 5e(yp) is called an "(L, ~ , Zl, n)-standard" MHI-system.

We can now state and prove the standard representation theorem.

T H E O R E M 1. Let L be a (O, I) lattice and ~ be a quasivariety of algebras of type ~. Suppose S c ' F { A } 2 x L is an (L, 3V, zi, n)-standard MHI-system, for some z l c ~ { B } E x L and n ~ 2 . Then V s = ~ { A } / { O } t is an algebra of "F, As:L--* Con (Vs) such that As(X)= { x } t / { O } # is a (0 , I) lattice homomorphism, and As[L] is an n-permutable sublattice of Con(Vs). Furthermore, the following are equivalent:

(1) There exists a (0 , I) lattice embedding )t :L ~ Con (V) for some V in ~" such that A[L] is an n-permutable sublattice of Con (V) and there exists a "r-homomorphism h:'V{B}--> V such that (h(f) , h(g)) is in X(x) for all (f, g, x} in zi.

(2) As is a (O, I) lattice embedding. (3) For each x in L, the polar {(a,, b,)}* with respect to S is the principal dual

ideal [x, I] of L.

Proof. Assume the hypotheses. Then As is the composition of the meet homomorphism x ~ {x}t preserving O and I from L into the principal dual ideal [{O}t, I] of Con(T'{A}), and the (O, I) lattice isomorphism 0~--> 0/{O}t from [{O}t, I] onto Con (Vs) (see [3: Thm. 3, p. 61]). A straightforward analysis of the transfinite construction shows that there are no faults or defects for the limit

sys tem S. Since S has no faults, Vs is in T'. Since S has no defects, {x v y}t = ({x}t{y}t{x}t.-.) , for all x, y in L. It then follows that As is a (O, I ) lattice homomorphism such that As[L] is n-permutable, proving the first part of the theorem.

Assume condition (1). Suppose (a~,bx, y) is in S and (u, v) is any pair in h (x )c V 2. We shall construct a l--homomorphism s : F{A}---> V with s(ax) = u and s(b~) = v, and such that (f, g, z) in S implies that (s(f), s(g)) is in h(z). Assuming this lemma, we have (u ,v )=(s(a , , ) , s (b , : ) )~h(y) , and so h ( x ) c h ( y ) . But then x c y since h is an embedding, and so {(ax, bx)} *C Ix, I]. Therefore, the lemma implies condition (3), since {(a~, b~)}* is a dual ideal of L containing x, hence must equal [x, I]. To define s, we define s(a) for a in Av by transfinite induction for 0<--_ u<--p, and show that any s~:~V{A}---~ V such that s~(a)= s(a) for all a in A~ has the property that (sv(f), sv(g)) is in h(z) whenever (f, g, z) is in 5~(Y~). For any ~--homomorphism t:~V{A}---> V, let S(t) denote the set of triples ( f ,g ,z) in


T'{A}2• such that (t(f), t(g)) is in A(z); it is easily checked that S(t) is an MHI-system. To begin the induction, define s(a) for a in Ao by s(ax)= u, s(b,) = v, s(az) = s(bz) = u for z # x in L, and s(b) = h(b) for b in B. If so : J/'{A}---~

V satisfies so(a) = s(a) for a in Ao, then 5e(Y o) c S(so) by the hypotheses for (u, v) and h, since S(so) is an MHI-system. So, assume the induction hypothesis. If u is a limit ordinal, then s(a) has been defined for all a in Av, and the induction step follows immediately. For a successor ordinal v + 1, u < O, suppose first that Tv was defined using a defect (f, g, x, y) for 5e(y~) such that f and g have support on A~. Since s is defined on Av, hence on T'{A~} by extending to a 1--homomorphism, we can obtain s(f) and s(g) in V. By the induction hypothesis, (f, g, x v y ) in 9~(Y~) implies that (s(f), s(g)) is in A(xvy) . But then ( s ( f ) , s (g) ) is in (A(x)A(y)A(x) . - - ) , by (I), hence there exist Vo, V ~ , ' " , v , in V such that vo=s( f ) , v, = s(g), and (vi-t, vl) is in A(x) for i odd, A(y) for i even, i = 1, 2 . . . . . n. So, we define s(a,,) = vi for i = 1, 2 . . . . . n - 1. If T, is empty, define s(a,,,) arbitrarily, say s(ai~,) = u for i = 1, 2 . . . . . n - 1. For s~+l :T'{A}---~ V such that s,,+t(a)=s(a) for a in A~+l, we have Y,,cS(s~,§ by the induction hypothesis, and T, ,c S(sv+~) follows from the definition of the s(a,,). Finally, U,, c S(s,,+~) is showed using the induction hypothesis, the hypotheses for the (~, ~) used to define a nonempty U~, and the assumption that V is in T'. This proves 5e(Y~+~)c S(s,,+~) and completes the transfinite induction. This defines s on A, proving the lemma. Therefore, (1) implies (3).

Assume condition (3). For x, y in L ,{x} t={y} t implies that (a~, b~, y) and (ay, by, x) are in S, and so x c y and y c x by (3). Therefore, x ~ {x}t is one-one, and so As is one-one also. This proves that (3) implies (2).

To prove that (2) implies (1), take V to be V s, A to be As and h to be the restriction of the canonical map 7/'{A}--*T'{A}/{O}'~ to ~{B}. Then A C Y o c S yields the required property for h. So, (1), (2) and (3) are equivalent. [ ]

w The completeness theorem.

We first set up our system in full generality by the appropriate definitions. For many applications, a simplified system can be used.

D E F I N I T I O N 2. Let C = {q : i -- 0}, D = {d~ : i ->_ 0} and E = {el : i >_-_ 0} be de- numerable sets of variables. Let o- denote the algebraic type (2, 2, 0, 0 )=

(A ,V, O, I) for bounded lattices, and P(E, o'i the o--algebra of o--polynomials on E. Let G denote the set of "generic pair" variables, that is:

G = {a(x):xaP(E, tr)} U{b(x) :x~P(E, or)}.

Vol. 13, 1981 Logic for n-permutable congruences 2t5

Let X = G U C LI D, let z be an algebraic type, and let P(X, -r) be the r-algebra of r-polynomials on X. (The variables of C will correspond to elements of B c A0 in

s tandard systems, and those of D will correspond to elements of B or to generators a~v used to correct defects.) For Z c X, let W(Z) denote the set of all triples in P(Z, z ) 2 x P ( E , tr); we will consider W(X) , W(C), W ( G t iC ) , etc.

A "r-system of hypotheses" H is a quintuple:

(~(0-), R(0-), X(z), A, n)

such that _V(o-) is a set of identities or implicational formulas for 0--algebras which contains the bounded lattice axioms, R(tr) c P(E, 0") 2, -v(z) is a set of identities or implicational sentences for z-algebras, A c W(C), and n>=2. (That is, V(o-) contains the commutativity and associativity laws for meet and join, the absorp- tion laws, and the identities a/x O = O and a v I = I, and may also contain other identities or implicational formulas for 0--algebras.)

DEFINITION 3. A "model" (V, a, L,/3, h) for a z-system of hypotheses

H =.(X(tr), R(0-), v(z) , A, n)

is given by diagrams:

P ( X , r ) '~ , V P(E, or) ~ ~ L ~ ~Co n ( V)

such that V is a -r-algebra, a is a z-homomorphism, L is a o--algebra, /3 is a (r-homomorphism, h is a o--embedding such that AlL] is an n-permutable sublattice of Con (V), L ~X(0-) (that is, every formula of v(0-) is satisfied in L), /3(x) =/3(y) for all (x, y} in R(cr), V~Y,(7), (a( f ) , a(g)> is in h~(x) for all (f, g, x) in zl, and (aa(y) , ab(y)} is in h/3(y) for all y in P(E, 0-). Note that L is necessarily a (O, I) lattice.

Assume that the terms of the formulas in _v(tr) are in P(E, or). A o-- homomorphism ~:P(E, tr)---~ P(E, or) is a substitution process assigning lattice polynomials to variables of E, so that L ~(p = q) for a lattice identity p = q in v(0-) implies that /3~(p)---/3~(q). Note that any particular such application is a finite process, since we only need to know the values.of ~ at the finitely many variables of p = q. Similar remarks apply to implicational formulas in X(tr). For s we assume that all formula terms are in P(C, z), and consider z-homomorphisms rl:P(C, z) ~ P(X, z).


D E F I N I T I O N 4. Suppose ~- is an algebraic type and

H = (-V(o'), R(o'), X('r), A, n)

is a ,r-system of hypotheses. We now define concepts of well-formed formulas, axioms, rules of inference, proofs and derivable formulas, which are appropriate for the logical system based on H.

The well-formed formulas of our theory are precisely the triples (f, g, x) of W(X).

Let A x ( H ) c W(X) denote the set of axioms for H. These axioms fall into seven schemes shown below, where x and y are any elements of P(E, o-) and f and g are any elements of P(X, ,r):

(A1) if, g, I) (Axioms for L) (A2) if,/ , x) (Axioms for reflexivity.) (A3) (a(x), b(x), x) (Axioms for generic pairs.) (A4) (Axioms for tr-identities.) For any o--identity p = q in ~(o-) and any

o--homomorphism r : P(E, o') ~ P(E, tr) :(a~(p), bE(p), ~(q)) and (a~(q), b~(q), ~(p)).

(A5) (Axioms for o--relations.) If (x, y) is in R(o'):

(a(x), b(x), y) and (a(y), b(y), x).

(A6) (Axioms for ,r-identities.) For any ,r-identity u = v in v( 'r) and any ,r-homomorphism rl: P(C, "r) ~ P(X, ,r): (r~(u), Tl(v), O).

(A7) (/, g, x) in A c W(C) (Direct axioms.) This completes the list of axiom types, defining Ax(H).

We give ten rules of inference for our logic, listing premisses above and consequences below a horizontal bar as usual. (Our rules may have more than one consequence.) Suppose x and y are any elements of P(E, tr) and f, g and h are any elements of P(X, ,r) below:

(f, g, x), (jr, g, y) (R1) (Meet rule for dual ideals.)

(f, g, x/x y)

(~, g, x) (R2) (Inclusion rule for dual ideals.)

(f, g, x v y )

(f, g, x) (R3) - - (Symmetry rule.)

(g, f, x)

(f, g, x), (g, h, x) (R4) (Transitivity rule.)

(f, h,x)

Vol. 13, 1981 Logic for n-permutable congruences

(R5) (Congruence property rule.) For any "r-polynomial and/~, gi in P(X, "r) for i = 1, 2 . . . . . k :

(ft, g,, x), (f2, g2, x) . . . . . (fk, gk, x)

(P(f, , f2 . . . . . fk), P(g,, g2 . . . . . gk), x)

(R6)

(R7)

(R8)

(a(x), b(x), y) (a(x v y), b(x v y), y)

(Join rule for generic pairs.)

( a (x) ,b (x) ,y ) , ( f i g,x)

(f, g, Y) (Substitution rule for inclusion.)

(Rule for implicational or-sentences.) For any sentence

(Pl = qt&'P2 = q 2 & ' ' ' & P ~ - I = q=-I):::)'P~ = qm

in X(o-) and any o--homomorphism ~: P(E, t r ) ~ P(E, o-):

(a~(p~). b~(p~), ~(q~)), (a~(q~), b~(qi), ~(p~)) for i -- 1, 2 . . . . . m - 1

217

p(al , a 2 . . . . . ak)

(a~:(p~), b~(p,.), ~(q,.)), (a~(qm), b~(qm), ~(Pm))

(R9) (Rule for implicational r-sentences.) For any sentence

(u~ = v2 & u2-- v2 & �9 " �9 & u , . _~ = v~ . -1 ) ~ u , . -- v . ,

in Z(r) and any r-homomorphism ~:P(C, r)--~ P(X, r):

(~(u~), "O(vi), O) for i = 1, 2 . . . . . m - 1

(n(u~), n(v~), o)

(R10) (Rule for n-permutability.) For n - 1 distinct unused variables dj,,di2, . . . . d i .... of D:

(f, g, x v y )

(f, dj,, x), (d h, dj2, y), (d,~_, dj~, x) . . . . . (d~._,, g, z)

where z = x for n odd, y for n even. A variable di of D is "unused" if the sequence of triples to which (RI0) is to be applied doesn' t contain any term having an occurrence of dv

This completes the list of rules of inference. We now define an "H-p roo f " as a finite sequence w = (w~, w2 . . . . . w,) of triples in W ( X ) by recursion as usual.


Starting f rom an axiom, we can extend any H-proof by adding any axiom. If an

H-p roo f contains all the premisses of a rule of inference, we can add the consequences of the rule to the end of the H-proof in any order. To make this precise, we use pairs (w, F) such that w is an H-proof and F is the set of rule C variables introduced by applications of (R10) in its proof. So, an " H - construction" for w = (w~, w 2 . . . . . w~) is a sequence of pairs:

((Wl, F0, (w2, F z ) . . . . . (w,, F,)),

with the following properties: w = w , , w t = ( w l ) for w~ in Ax(H) and F~ = (empty set), and for l < ] < t , wi_~=(wl , w2 . . . . . ws) for some s<r. and either

w i = (wl. w2 . . . . . ws+l) for ws+l in Ax(H) and F~ = Fj_ 1, or {wl. w2 . . . . . ws} contains all the premisses for some rule of inference (R1) through (R9), w i = (wt, w 2 . . . . . w~+k) where {w~+,, w~+2 . . . . . ws+k} is the set of k consequences of that rule and Fj =Fi_ ~, or w i = ( w l , w2 . . . . . w~+.) and:

6 =6_ , u{4, , 4:, . . . . dj. ,}.

where w i is obtained by applying (R10) to some wk for k<-_s and using di, in D not occurring in Wi_l. Define w to be an H-proof if there is an H-construct ion (w i, F~) for j~=t such that w = w , .

We say that (f, g, x) in W(X) is "semiderivable from H " if it is the last t r ipleof some H-proof . If (f, g, x) is semiderivable from I t and f and g contain no rule C variables from applications of (Rl0) in some H-p roo f for it, then (f, g, x) is called "der ivable" from It . More precisely, (f, g, x) is derivable f rom H if there exists an H-construction (w i, F i) for ] N t such that (f, g, x) is the last te rm of w, and no variable d, in F, occurs in f or in g. Note that semiderivable triples in W(G LJ C) are derivable, since they contain no variables d~. This concludes the formal definitions, and we can now state the completeness theorem.

T H E O R E M 2. Suppose "r is an algebraic type and

I t = (Y.(o-), R (o-), v (.r), A n)

is a T-system of hypotheses. Then the following statements hold for all x, y in P(E, o') and all f, g in P(X, "r):

(1) a(f) = a(g) for all models (V, a, L, [3, )Q of H if and only if (f, g, O} is derivable from H.

(2) [3(x) = [3(y) for all models (V, a, L, [3, ~) of H if and only if (a(x), b(x), y) is derivable from H.


(3) (a( f ) , a(g) )~ h/3(x) for all models (V, a, L, ~, h) of H if and only if (f, g, x}

is derivable from H.

Proof. Assume the hypotheses. We prove correctness by induction on the length of H-constructions, using the following formulation:

Suppose (V, a, L,/3;,k) is a model for H and (wi, F/} for j = 1, 2 . . . . . t is an H-construction, with:

W, = (1"71, N 2 . . . . . Wr} for wi = if~, g~, x,), i = 1, 2 . . . . . r.

Then (C,) there exists a z-homomorphism a ' : P(X, "r) --+ V with a' (x) = a(x) for all x in X - F , such that (a'(f~),a'(g~)) is in h/3(x~) for i = 1,2 . . . . . r, and (D,)/3(y)c/3(Xk) if fk =a(y ) and gk =b(y ) for some y in P(E, o') and k<=r.

For each t>_-l, (Ct) implies (/9,). Suppose fk = a(y) and gk = b ( y ) for some k - r . Given (u, v) in h/3(y), there is a 1"-homomorphism ao:P(X, "r)--+ V such that ao(a(y))= u, ao(b(y))= v, and ao(x)= a(x) for x in X -{ a (y ) , b (y )} . Then (V, ao, L,/3, h) is a model for H, and so by (C,) there exists a ' such that:

iu, v ) = ia,,a(y), aob(y))= ia'( fk), a ' (gk))c h/3(xk).

But then h/3(y)= h/3(xk), and so /3(y)c/3(xk) because h is an embedding. This proves that (C,) implies (D,).

Now (C~) holds by direct calculation, using a '= a. Suppose t > 1 and (C,-0 holds, and so (D,_0 holds by the above. Again, direct calculation shows that if w, is obtained from w,_l by adding an axiom or using one of the rules of inference (RI)-(Rg), then (C,) follows using a ' = a", where a" comes from (C,_0 applied to w,_ L. If w, is obtained from w,_, using (RI0), then we have w i = if, g, x v y ) for some j _-< r - n, and

{wr_,+l, wr-,,+ a . . . . . w,} = {iui-1, ui, zi) : i = 1, 2 . . . . . n},

where u, = di, (unused in w,_ 1) for i = 1, 2 . . . . . n - 1,

uo = f, ~ = g and z~ is x for i odd, y otherwise.

Since h[L] is n-permutable, and ia"(f) , a"(g)) is in h/3(x v y) by (Ct_,), there exist V0, Vl . . . . . "/')n in V such that (v~-l, vi) is in h/3(zi) for i <- n, v0 = a"(f) and v, = a"(g). Defining a ' to be the ~--homomorphism such that a'(u~) = vi for 1 --U_ i ~ n - 1 and a'(x) = a"(x) for other x in X, the induction to (G) follows.

If if, g, x) is derivable from H, we have:

(a( f ) , a ( g ) ) = (a ' ( f ) , a ' (g) )~ h/3 (x),


using an appropriate (C,) and the definitions. In particular, if x = O, then )q3(x) = O and so ct(f) = or(g). By an appropriate (D,),/3(x) c/3(y) if (a(x), b(x), y) is derivable from I-I. This proves the correctness of the logic for I-I.

To prove the completeness of our theory, we use the triples derivable from I-I to construct a standard MHI-system that yields an appropriate model for I-I. First, we recover L and /3 by constructing a congruence on P(E, o-). Let 0(I-I) denote the set of all pairs (x, y) in P(E, o-) 2 such that (a(x), b(x), y) and (a(y), b(y), x) are both derivable from I-I. Now, 0(H) is an equivalence relation by (A3) and (R7). Supposing that (a(xt), b(xt), Yl) and (a(x2), b(x2), Y2) are derivable from H, then so are the following:

(a(x, A x2), b(x,

(a(x, A x2), b(x,

(a(x,/x x2), b(x,

(a(xl/x x2), b(Xl

(a(x~ A x2), b(x,

^ x~), (x, ^ x2) v xl)

A x2), x0

Ax2), y,)

A x2), y25

A x2), Yl A Y25

by (A3) and (R2),

by (A4) and (R7) several times,

by (R7) and the hypotheses,

by steps similar to the above, and

finally, by (R1).

It follows directly that 0(I-I) is a meet congruence. A similar argument, requiring (R6), shows that 0(H) is a join congruence, and so is a o--congruence. Define L to be the o--algebra P(E, o-)/0(H) and /3:P(E, o-)---,L to be the canonical o-- homomorphism. By (A4) and (R8) we have L g ~(o-), and so L is a (O, I) lattice in particular. From (A5), we see that /3(x)=/3(y) for all (x, y) in R(o-). Also, /3(x)c f3(y) if and only if (a(x), b(x), y) is derivable from H, by an easy computa- tion which we omit.

For a sufficiently large infinite cardinal p, take as usual:

A = AoU{aiv: 1 <=iNn- 1, u < 0 }

for a standard MHI-system on T'{A}axL. Here, A o = X = G U C U D , the r- algebravariables. For genericpair variables in Ao, leta~ = a(~) andb~ = b02), choosing

in P(E,O-) such that /3(~)=x for each x in L. Then we let A o = X = H U B , where:

H={a , , : x~L}U{b , , : xcL} and B = X - H .

Let T" be the quasivariety of r-algebras axiomatized by v( r ) . Let ~ :P(X, r ) -+ T'{A} be the unique r-homomorphism leaving each element of X fixed, that is, with cta(y)=a(y) and a b ( y ) = b(y) for y in P(E, o-) and ct(ci)= ci and o~(~)=


for i-->--0. Using Ll and n from H, we construct an (L ,~ ,za(H) ,n) -s tandard MHI-system S = S(H), defining ~I(H)= ( K o - K ) U A o as given below:

K={(a~,b~,x):x~L}, Ko={(aa(y),ab(y),/3(y)):y~P(E, cr)}

and

,a,, = { ( a ( f ) , a ( g ) , / 3 ( x ) ) : ( f , g, x) ~ ~ }.

Then S =b~ for {Y,,},,<--o given by:

Y. = K U A(H) = Ko U Ao. Y~= U Y~ for limit v,

and

Y~+~=Y~UT~UU~ for v < p ,

where T, and U~ are defined with respect to well-orderings of T and U (possible defects and faults), as previously described. Let V s = T'{A}/{O}t, so Vs is in T" and As : L --~ Con (Vs) given by As(X) = {x}t/{O}t is a o--homomorphism such that As[L] is an n-permutable sublattice of Con(Vs), by Theorem 1. Let as :P(X, ~)---~ Vs be the ~--homomorphism 8a, where 8 is the canonical ~-- homomorphism V{A}--> Y{A}/{O}t. For (f, g, x) in W(X), (a(f), a(g),/3(x)) is in S and if and only if (as(f), as(g)) is in hs/3(x). So, Ao~ Yo = S implies that (as(f), as(g)) is in hs/3(x) for all (f, g, x) in A. Similarly, Ko c Yo = S implies that (asa(y), ash(y)) is in hs/3(y) for all y in P(E, or). It only remains to prove that As is one-one to show that (Vs, as, L, /3, As) is a model for H.

We intend to prove s where ~ is the set of triples in W(X) that are derivable from H and s is the set of all (f, g,x) in W(X) such that (aft), a(g),/3(x)) is in S. (In fact, ~ = ~ . ) Suppose ~ ' c ~ and x and y are in L such that (as, b~, y) is in S. Since (as, b~, y )= (aa (2 ) , ab(2), /3(~)), (a(i) , b(i) , ~) is derivable from H because ~ c ~ , and so x =/3(2)c/3(~) = y. For x in L, this implies {(ax, b~)}* c [x, I], and so {(ax, bx)}* =[x, I] because {(a~, b~)}* is a dual ideal of L containing x. It follows by Thm. 1 that As is an embedding, and so (Vs, as, L,/3, As) is a model for H. If (f, g, x) in W(X) is not derivable from H, then it is not' in ~, and so (as(f), as(g)) is not in hs/3(x). Also, as(f)r as(g) if (f, g, O) is not derivable from H. We noted before that /3(x)=/3(y) fails if (a(x), b(x), y) is not derivable from H. Therefore, the completeness of the logic for H follows from ~ c 9.


To avoid conflicts of rule C variables, certain restriction and renumbering methods for variables of D in H-proofs are used. We will consider W(XN) for the subset XN = G U C U {dl : i <- N} of X. If (wj, F/) for j =< t is an H-construct ion such that F, = X - X N = {dl : i > N}, then w, is called an "N-res t r ic ted" H-proof . Sup- pose w is an H-proof , and we fix a particular H-construction (w i, F/) for ] -< k such that w = wk. Let @(w) denote the set of triples Wo in W ( X ) such that there is an H-construction (w i, F~) for ]-< t (k < t) extending the H-construct ion for w and such that Wo is the last term of w, and no d~ in F, - F k occurs in Wo. (That is, Wo is semiderivable by extending w and Wo may contain rule C variables of w, but not rule C variables used in extending w to Wr) Clearly, any term of w is in @(w). Also, @ = ~(w). (If w' is an H-proof and the last term Wo of w' doesn' t contain any rule C variables of w', then construct w" by replacing each rule C variable d, in w' by dN+i. For large enough N, the concatentation of w and w" is an H-proof with last term Wo, proving Wo~@(w).) For w N-restricted and any finite J c @ ( w ) , we can prove similarly that there exists an N-restricted H-proof w' = (wz, w2 . . . . . ws) extending w such that J = {wl : i <= s}. So, if @(w) contains the premisses of any rule (R1)-(R9), then it also contains the consequences.

For any ~:P(X,'r)-->'I/'{A}, we define (K2•215 by (KZ• g ,x) )=(K(f ) , K(g),/3(x)). (So, ~" is the inverse image (c~2x/3)-l[S].) For N>--0, we call (K,w) an "N-pa i r " if w is an N-restr icted H-proof and K : P(X, -r) --> ~{A} is a r -homomorphism such that K(x) = o~(x) = x if x is in XN, K [ X ] = A , and K restricted to X is one-one, and so has a partial reciprocal K-l: K[X]---, X.

Suppose N=>0 and (K,w) is an N-pair. Then an N-pair (3,,w) is an " N - variant" of (K, w) if 3'(4) = K(di) for i = 1, 2 . . . . . M, where M>_- N such that every triple of w is in W(XM). Define:

~(~, w) = (K2 x t3)[~(w)] = {(~(f), ~(g), t3(x)) :(f, g, x) ~ ~(w)},

~(K, w) = U {M(3', w) : (% w) is an N-variant of (K, w)}.

Some useful properties of these constructions are given below.

L E M M A 1. If (K2x/3)(Wo)~ ~(K, W)for WO in W(X) , then WoS@(w).

L E M M A 2. I f (%w) is an N-variant of (~:,w) such that K[X]= 3,[X], then s~( K, w) c s~(,/, w).

L E M M A 3. If J is a finite subset of ~(~r w), then there is an N-variant (% w) of ( Jc, w) such that J c sl( % w).

L E M M A 4. ~(K, w) is an MHI-system on ~{A} 2 x L.

Vol. 13, 1981 Logic for n - p e r m u t a b l e congruences 223

For Lemma 1, we suppose that wo=(f , g, x) is in W(X) such that (KS• is in si(•,w), sO (~:2•215 for some wl=(u,v , y) in ~(w). Let r denote the set of pairs (f', g') in P(X, "r) 2 such that (f ' , g', O) is in 3 . By (A2), (R3), (R4) and (R5), t0(H) is a -r-congruence on P(X, "r). So, let V = P(X, ,r)/to(H), and let ~ :P(X, "r)--~ V be the canonical ~--homomorphism. By (A6) and (R9), V is in 7/'. Let ~ :7/'{A} ~ V be any "r-homomorphism such that ~(a) = "trK-l(a) for each a in ~:[X]. Then ~K(x)= ~'K-1K(X)= lr(x) for all x in X, so ~K =rr . Now K(u)= K(f) by the above, so zr(u)=~(u)=~K(f)=Tr(f) , and therefore (u,f, O ) ~ J . Similarly, (v, g, O) is in ~. Also, (a(O), b ( O ) , x ) and (a(y), b(y), x) are in ~ because {3(O) c /3 (x) =/3(y). But then (u, v, x), (u, f, x) and (v, g, x) are in ~(w) by (R7), and so wo = (f, g, x) is in ~(w) by (R3) and (R4). This proves Lemma 1.

Assume the hypotheses for Lemma 2. Suppose z ~ M(K,w), so there is an H-construction (wi, F/) for j<=t such that Wt-~(Wl, W2 . . . . . Ws) with z = (~r215 W=Wk=(Wl, W2 . . . . . Wr) for some k < t and r<s, and no di in F, -Fk occurs in w~. Define ~:P(X,'r)--->P(X,'r) to be the unique -r- homomorphism such that ~(x)= 3~-l(K(x)) for x in X, so V~(x)= 3,~/-1K(x)= K(x) for x in X, and hence 3'~ = K. For i = 1, 2 . . . . . s, suppose w~ = (f~, g~, x~), and define w~=(~(fl), t(gi), x~). By direct calculation, w'=(w'~, w ~ , . . . , w~') is an H-proof. Since wl, wz . . . . . w~ are in W(X~) for some M such that ~/(d~) = r(d~) for i <_- M, we have w" = w~ for i -<_ r. It follows that w" is in ~(w) via w'. Using ~/~ = K, we have z = (~/2 • ~ s~(V, w), proving Lemma 2.

Lemma 3 follows from Lemma 2, and Lemma 4 follows from Lemmas 1 and 3, (A1), (A2), and (R1)-(R5). We omit these computations.

We now prove by transfinite induction for v<-p the statement: (Q~). If N->_0 and J is a finite subset of 9~ then there exists an N-pair (K,w) such that J c sg(K, w). Let wo be any N-restricted H-proof, and observe that Yo C ~ (a , Wo) by (A3) and (A7), so 5e(Y o) ~ ~(c~, Wo) by Lemma 4. Then (Qo) follows by Lemma 3. If v <- O is a limit ordinal, then (Q,) follows from the induction hypothesis (Q, ) for all ~ < u because 5e is algebraic. So, it remains to prove (Q,+x) for the successor ordinal v + 1, assuming the induction hypothesis (Q,) for v < p. Suppose that T~ is defined from some defect (f, g, x, y) for 9~ and U~ is defined from some fault (w, n) for b~ where ~ is given by:

( U l = Vl & IA2 ~--- I)2 & " " " & [~a--1 = ' 0 m - - 1 ) ~ IAm = 1)m-

Let Y be a finite subset of 5P(Y,+~)=Se(Y~t.JT~tAU,), and observe that J = 5~ U T, tA U~) for some finite subset Jo of Y~, since 5r is algebraic. Now define the finite subset Y~ of 9~ as follows:

J~ = Yo U {(f, g, x v y)} U {(-q(u~), n(v,), O) : i = 1, 2 , . . . , m - 1},

and note that J~ r 7/'{A~}~ • L by the definitions of Y~, T~ and U~. By (Q,), there


exists an N-pa i r (K1, Wl) such that Jz c ~(Ka, Wl). Choose a finite subset A ' of A~ such that J1 c 7/'{A'}2 x L and rl(ci) is in 7/'{A'} for every variable c i occurring in

_~. Construct an N-var ian t (K2, wl) of (K1, wz) such that KI[X]t.JA'c K2[X], so 11c~/(K2, w1) by L e m m a 2. Let X ' = K 2 1 [ A ' ] c X , so that for any u in 7/'{A'} there exists fi in P(X ' , 1-) such that K2(fi) = U. SO, there exists a "r-homomorphism

7] :P(C, 1")~ P(X, T) such that "~(c i) is in P(X' , -r) and K2~(c i) = rl(ci) for every variable cj in ~ . Then K2~](Ui)=~I(Ui) and K2~(vl)='q(vi) for i = 1 ,2 . . . . . m. Recall that /3(~)= z for z in L. By L e m m a 1, we see that ~(wl) contains (1)

(fi, ~7, ~) for all (u, v, z) in Jo, (2) (fi(u~), ~](vi), O) for i = 1, 2 . . . . . m - 1, and (3) the triple (f, g, ~ v g ) . So, we can construct an N-pa i r (K2, W2) where wz = (Wl, w2 . . . . . w~) extends wz and each triple of (1), (2) and (3) above is in {wi :i>=s}. By (Rg), we can suppose that w~ =(fi(um), ~](v,,), O). Choose M>=N

large enough so that every term of w2 is in W(XM) and d~ in X ' implies that i<=M. Let w3 be obtained from w 2 by applying (RI0) to (f, g, x x/9), employing the unused variables dM+z, dM+2 . . . . . dM+,-1. Since A ' and A , + I - A ~ are dis-

joint, we can construct an N-pai r (K3, w3) such that K3(~) = x2(d~) for dl in X ' and Ka(d~+~)=a~ for i = 1 , 2 . . . . . n - -1 . I t follows that J o k J T ~ U U ~ s g ( K 3 , W3), hence Jeff,(K3, w3) by Lemma 4, hence J c sg(K, w) for some N-pa i r (K, w) by L e m m a 3. So, (Q~) implies (Q~+I) when T~ and U~ are nonempty, and the other cases are proved similarly. This completes the transfinite induction.

Suppose Wo ~ g, so Zo = (a 2 x/3)(Wo) ~ S = 5e(%). Choose N_-> 0 so that Wo W(XN), and construct an N-pa i r (K, W) such that Zo6 ~(K, w) using (Qo). Since K equals ot on P(XN, "r), we have Zo=(K2x[3)(Wo), and so W o ~ ( w ) by L e m m a 1. Since w is N-restr icted, Wo is derivable from t t by a suitable extension of w. Therefore r c ~ , completing the proof of Theorem 2. []

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A complete logic forn-permutable congruence lattices

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