Algebra Universalis, 34 (1995) 404-423
0002-5240/95/030404-20501.50
+ 0.20/0
9 1995 Birkh/iuser Verlag, Basel
Total tense algebras and symmetric semiassociative relation algebras
P.
JIPSEN, R.
L. KRAMER AND R. D. MADDUX
Abstract. It is well known that the lattice ARA of varieties of relation algebras has exactly three atoms.
An unsolved problem, posed by B. J6nsson, is to determine the varieties of height two in ARA.
This paper solves the corresponding question for varieties generated by total tense algebras. More
specifically,we show that there are exactly four finitelygenerated varieties and infinitelymany nonfinitely
generated varieties of height two. In the second half of the paper we show that total tense algebras are
term equivalent to certain generalized relation algebras and extend our results to varieties of these
algebras.
1. Introduction and definitions
Tense algebras are the algebraic c o u n t e r p a r t of tense logic, which has been
studied a n d used by m o d a l logicians to reason a b o u t tenses in a language a n d a b o u t
t e m p o r a l processes. We first describe a typical tense algebra.
Let U be a set a n d R a b i n a r y relation o n U. The relation R gives rise to two
u n a r y operations f a n d g on the set of all subsets of U, n a m e l y
f ( X ) = the image of X u n d e r R
= {u 9 U : x R u for some x 9 X}
g(J() = the preimage of X u n d e r R
= {u 9 U : u R x for some x 9 X}.
If U is interpreted as a set of events,
between events in U, then f ( J ( ) is the
X, a n d g ( X ) is the set of events in the
o b t a i n e d if the operations f a n d g are
a n d R is interpreted as a t e m p o r a l relation
set of events in the future of some event in
past of some event in X. A tense algebra is
added to the Boolean algebra of all subsets
Presented by B. J6nsson.
Received April 8, 1994; accepted in final form December 1, 1994.
404
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
405
of U. The resulting tense algebra is the complex algebra of ~ U, R ) , namely,
era(u, R) =
(Sb(U), u,
,f, g).
It follows from the Extension Theorem of J6nsson and Tarski [7], Theorem 2.15,
that every tense algebra can be embedded in one that arises in this concrete fashion.
The abstract definition of a tense algebra runs as follows. A tense algebra is an
algebra of the form 9,I = (A, + , - , f , g ) , where
(i) (A, + , ) is a Boolean algebra and
(ii) f and g are conjugate operations, i.e., for all x, y E A
y .f(x) = 0
iff x .g(y) = 0.
Let N be a tense algebra. Define the binary operation - by x . y = 2 + )5 for all
x and y. By (i) we may define the distinguished elements 1 and 0 by 1 = x + )? and
0~1.
Property (ii) is quite strong since it implies that f and g are additive
( f ( x + y) = f ( x ) + f ( y ) and g(x + y) = g(x) + g(y)), normal ( f ( 0 ) = 0 = g(0)), and
monotone (if x _<y then f ( x ) < _ f ( y ) and g ( x ) < g ( y ) ) . By Huntington [3], [4] and
J6nsson-Tarski [7], Theorem 1.15, both (i) and (ii) can be expressed by the
equations listed below, so the class of all tense algebras is a variety.
x + ( y +z) =(x +y) + z
x +y =y +x
x=2+)5+2+y
f(O) = 0
g(0) = 0
f ( x ) "y < f ( x "g ( y ) )
g ( y ) " x <_g ( y . f ( x ) ) .
Furthermore, the last four equations can be replaced by the following two:
f(x
g ( y ) ) <_f(x) " ~
g ( y "f ( x ) ) <_g ( y ) " 2.
406
P. J1PSEN ET A L.
ALGEBRA UNIV.
A tense algebra is called reflexive if it satisfies the identity x <_f(x). An equivalent
identity is x <_ g(x). To see that these two identities are equivalent, assume x <_f(x)
for every x. Then x . g ~ < _ f ( x . g(x)), so x . g ~ = x . g(x) . f ( x . g(x)). But x g(x) . f ( x . g(x)) <_ x . f ( x . g(x)) _<_x . f ( x ) . 2 = 0, hence x . g(x) = 0, i.e., x < g(x).
Similarly, if 91 satisfies x <_g(x), then it also satisfies x <_f(x). A tense algebra is
called t o t a l i f i t satisfies the implication x # 0 ~ f ( x ) + g(x) = 1. Of course a binary
relation R on a set U is reflexive if x R x for every x ~ U, and total if x R y or y R x
for all x, y e U. (Note that a total relation is reflexive.) Not surprisingly such
relations give rise to correspondingly named tense algebras, that is, R is reflexive on
U if and only if ~ m ( & , R ) is a reflexive tense algebra, and R is total on U if and
only if (!;re(U, R ) is a total tense algebra.
The a t o m structure of a tense algebra 9I is the structure At(91) = ( 5 7, R ) , where
U is the set of atoms of 91 and R = {(u, v) E U x U :f(u) _> v}. If 91 is complete
and atomic then 91 is isomorphic to the complex algebra of its atom structure.
Finite tense algebras are complete and atomic, so they are always isomorphic to the
complex algebras of their atom structures. Figures 1 and 2 show the atom structures
of the ten nonisomorphic total tense algebras with at most 3 atoms and at most 8
elements. The dots and circles represent atoms. Two atoms u and v are connected
by a single-headed arrow from u to v if the pair (u, v) is in R. A double-headed
arrow between u and v means that both (u, v) and (v, u ) are in R. The ten atom
structures in Figures 1 and 2 arise by enumerating all total relations on sets with 1,
2 or 3 elements.
At(To)
At(T1)
At(T2)
/02
/~
At(T3)
At(T4)
Figure 1. Necessary subalgebras of finite total tense algebras.
At(Ts)
/\
/\
/\
/\
At(T6)
At(TT)
At(Ts)
At(T9)
Figure 2. Remaining total tense algebras with 8 elements.
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
407
Every total tense algebra is a discriminator algebra, and hence is simple. The
variety generated by all total tense algebras is finitely based. It is not hard to show
that a basis is given by the equations above together with x + f ( f ( x ) + g(x)) +
g ( f ( x ) + g(x)) <_f(x) + g(x). See Jipsen [5] for details.
Finally, we note that if 9.1 is the complex algebra of the structure { U, R ) , then
every complete subalgebra of 9/corresponds to a partition of the set U with the
property that the image of each block of the partition under f and g is a union of
blocks. The blocks are the atoms of the subalgebra. This observation will be very
heavily used in the proof of Theorem 1 below.
2. Total tense varieties of height 2
The lattice ATA of aU varieties of tense algebras has infinitely many atoms, but
only one of these atoms is a variety of reflexive tense algebras, namely the variety
generated by the tense algebra T o = f f m { { x } , { { x , x ) } ) . This follows from the
observation that every nontrivial reflexive tense algebra has a smallest subalgebra
isomorphic to T 0. We refer to this subalgebra as the constant subalgebra, since it
contains only the constants 0 and 1.
The variety Vat(T0) is in tuna covered by infinitely many varieties of reflexive
tense algebras. However we are interested in the varieties generated by total tense
algebras, since they are later shown to correspond to varieties generated by simple
subadditive semiassociative relation algebras. We now prove that only four of the
varieties covering Var(To) are finitely generated, i.e. generated by finite total tense
algebras. The atom structures of these four finite tense algebras are shown in
Figure 1.
T H E O R E M 1. I f R is a total relation on a set U and [U! >4, then ~ m ( U , R )
has a proper nonconstant subalgebra.
Proof. For any u, v G U, we say that "u points at v" just in case @, v) E R. The
hypothesis that R is total can then be expressed by stating that for any u, v e U,
either u points at v or v points at u. Since this holds even when u = v, it follows that
R is reflexive. Hence, for every X _: U,
f ( X ) D_X
and
g(X)~_X.
(1)
Suppose u e U and u Cf(X). Then no element o f X points at u, equivalently,
g({u}) c~X = ~ . Since R is total, u points at every element of t ; hence f ( {u }) __ X.
408
P, J I P S E N E T A L .
ALGEBRA
UNIV.
Summarizing, we have
if u Cf(X) then g({u}) c ~ X = ~ and f({u}) _~X.
(2)
By similar reasoning we also have
if u Cg(X) thenf({u})c~X = ~ and g({u}) _~ 2".
(3)
Choose any X __ U such that [X[ >_ 2 and X r U. Consider the partition
X and {u } for every u ~ U - 3(.
(4)
By the hypotheses on X and U, there are at least two sets in (4), and not all of them
are singletons. Consequently, if the sets in (4) form the atoms of a subalgebra of
Era(U, R ) then that subalgebra is both nonconstant and proper, hence we are
done, so assume otherwise. Since f(X) and g(X) both contain X, by (1), they are
clearly unions of sets in (4), so there must be some u s U - X such that either
f ( { u }) or g({u}) is not a union of sets in (4). Since X is the only nonsingleton in (4),
it follows that either X lies partly inside f({u}) and partly outside f ( { u }), or else X
lies partly inside g({u}) and partly outside g({u}). In the former case, the second
conclusion of (2) is false and the first conclusion of (3) is false, while in the latter
case, the first conclusion of (2) is false and the second conclusion of (3) is false. In
either case the hypotheses of (2) and (3) are false. Then u ef(X) c~g(X). Hence u
peints at some element x ~ X, and some y ~ X points at u. We argue next that we
may assume x and y are distinct. Suppose not. Since IX[ > 2 we may choose w e X
such that w r
= y . If w points at u, use w instead of y. I f u points at w, use w
instead of x. This shows that we may therefore assume
if X _= U, ]X[ > 2, and X r U, then these exist u e U - X and
distinct x, y ~ X such that u points at x and y points at u.
(5)
Let tl, Y _~ U. The pair (X, Y) will be called an T2-pair if the following conditions
hold: Xc~ Y = ~ , f ( X ) c~f(Y) c~g(X) c~g(Y) ~_X w Y, and ] X u Y] _> 3. I f (X, Y) is
a T2-pair, then X r ~ and Y r ~ , since otherwise ~ ~_ X u Y va ~ by the second
condition, and either X or Y contains at least two elements by the third condition.
The reason for the name is that if (X, Y) is a T2-pair then ~ m ( X w Y, R c~
((X u Y) x (X u Y))) is isomorphic to Ta.
The collection of T2-pairs is closed under unions of chains, in the following sense.
Let e be an ordinal and suppose (X~, Y~ ) is a T2-pair for every ~c < e. I f X~ _ X~
and Y~ _~ Y~ whenever ~c < Z < ~, then < U K< ~X~, U ~< ~ Y~ } is also a T2-pair.
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
409
Suppose that there is a T2-pair in (U, R ) . By Zorn's Lemma there must be
some maximal T2-pair (X, Y). If X w Y = U then Era(U, R ) has a nonconstant
proper subalgebra isomorphic to T2, as desired. On the other hand, if X w Y r U,
then we claim that ~ m ( U , R ) has a nonconstant proper subalgebra whose atoms
are
X, Y, and {u} f o r u e U - X - K
(6)
To prove this we need only check that the f-image of every set in (6) is a union of
sets in (6). Since (X, Y) is a T2-pair, f ( X ) contains both X and Y, but all other sets
in (6) are singletons, so f ( X ) is clearly a union of sets listed in (6). Similarly, f (Y),
g(X), and g ( Y ) are unions of sets in (6). Let u E U - X - Y. It suffices to show that
f({u}) either contains or is disjoint from X, and either contains or is disjoint from
Y. Since R is total and X ~ ~ , either u E f ( X ) or u ~ g(X). However, if u E f ( X ) c~
g(X), then (X, Y w {u}) is a T2-pair strictly larger than (X, Y), contradicting the
maximality of (X, Y). Therefore either u o f ( X ) - g(X) or u ~ g(X) - f ( X ) . If u
f ( X ) - g ( X ) then f({u}) ~ X = ~ and g({u}) ___X by (3), while if u E g(X) - f ( X ) ,
then g({u}) c ~ X = ~ and f({u}) _~ X by (2). In any case f({u}) and g({u}) either
contain or are disjoint from X. By similar reasoning we also conclude that f({u})
and g({u}) either contain or are disjoint from Y.
So far we have shown that if there is a T2-pair in (U, R ) , then Era(U, R ) has
a nonconstant proper subalgebra. We may therefore assume that
there is no T2-pair in (U, R).
(7)
This means, in particular, that there can be no "4-cycle" in R, that is, no four distinct
elements u~, u2, u3, u4 e U such that (ul, u2), (u2, u3), (u3, u4), (u4, ul ) e R, for
then ({ul, u3}, {u2, u4}) would be a T2-pair, contradicting (7).
Suppose there is a "3-cycle" in R, that is, three distinct elements ul, u2, u3 ~ U
such that uj points at u2, u2 points at u3, and u3 points at ul. Let X = {ul, u2, u3}.
By (5) there exist u ~ U - X
and distinct x, y e X , such that u points at x
and y points at u. Let z be the third element of X, distinct from x and y. Suppose
y points at x. X forms a 3-cycle under R, so x points at z and z points at y (see
Figure 3(i)). It follows that there is a 4-cycle, from u to x to z to y to u. The
existence of a 4-cycle contradicts (7), so we may assume x points at y. This implies
that the 3-cycte in X goes from x to y to z to x. There are two final cases. If u points
at z (see Figure 3(ii)), then there is a 4-cycle from u to z to x to y to u, contradicting
(7), but if z points at u (see Figure 3(iii)), then there is a 4-cycle from z to u to x
to y to z.
410
P. JIPSEN ET AL.
Xo
z0 -
4
0 I/
,.oy
(i)
0~o
zO=
~
--o
--oy
(ii)
ALGEBRA UNIV.
Xo
IL
9
zo
0 I/'
oy
(iii)
Figure 3
We may therefore assume that R contains no 3-cycle. Now suppose R contains
a 2-cycle, that is, two distinct elements us, u2 ~ U such that ul points at u; and u2
points at u t. By (5) there is some u s U - {ul, u2} which points at one of the
elements of {us, u2} and is pointed at by the other. Either way we get a 3-cycle. We
may consequently assume R has no 2-cycles, that is, R is antisymmetric.
So far we may assume R is reflexive, total, and antisymmetric. We may also
assume R is transitive, since the only way it can fail to be transitive is to contain a
3-cycle. Hence R is a linear ordering of U. Choose any u ~ U which is not a
minimum element under this ordering (where (u, v ) e R is interpreted as "u is less
than or equal to v"). T h e n f ( { u } ) # g and it is easily checked that the setsf({u})
and U - f ( { u } ) form the atoms of a nonconstant proper subalgebra of gin(U, R )
isomorphic to T~.
[]
C O R O L L A R Y 2. In the lattice of varieties generated by total tense algebras,
Var(Ti) ( i - - 1 , 2, 3, 4) are the only finitely generated covers of Var(To).
Proof Note that if 9.1 and ~ are two finite subdirectly irreducible algebras in a
congruence distributive variety then it follows from J6nsson's Lemma that Var(9.I)
is covered by Var(~3) if and only if 9g is isomorphic to a maximal proper subalgebra
of ~.
As mentioned in the introduction, a finite total tense algebra is isomorphic to
the complex algebra of its atom structure, so the previous theorem implies that
every finite total tense algebra with more than 8 elements has a proper nonconstant
subalgebra (with at most 8 dements), and hence cannot generate a variety that
covers Var(T0). Figures 1 and 2 show the atom structures of the ten nonisomorphic
total tense algebras with at most 8 elements. Note that Ts, T6 and T7 have a
subalgebra isomorphic to T I, whereas T 8, T9 have a subalgebra isomorphic to T2
(obtained by identifying the black vertices in the atom structures). On the other
hand TI, T2, T3 and T4 have no proper subalgebras other than the constant one,
hence they generate covers of Var(To).
[]
Vol. 34, 1995
Total tense algebras and symmetric semiassociativerelation algebras
411
3. A sequence of infinite total tense algebras
In this section we give countably many examples of nonfinitely generated
varieties of total tense algebras that cover Var(T0). These examples are based on
the veiled recession frame, used by W. Blok [2] to give similar examples of varieties
of modal algebras.
A structure (U, R ) is called an n-recessionframe if U = { a l , . . . , a, } and
R = {(ai, aj} : i >_j} w {(ae,
ai+1) : 1 <. i "< n}.
If we instead consider sets U = {as, az, a 3 , . . . } or U = {. . . . a_l, ao, ai, a2 . . . . }
with the natural extension of R then we obtain the co-recessionframe and the
Z-recession frame respectively. A veiled recession frame is the subalgebra of
fire(U, R ) generated by the singletons (or finite subsets) of Sb(U). This subalgebra
is denoted by ffmy(U, R). A diagram of the Z-recession frame is given in Figure
4(i). The downward pointing bold arrow represents the linear order part of R, and
is intended to indicate that, in addition to the arrows shown, there are also arrows
from every element to all elements below it (as well as to itself). Observe that
~ m ( U , R ) is a total tense algebra.
al,n
1
ao,n~
i
a-l,n~
,1
(i)
/
(ii)
Figure 4. Infinite total tense algebras with no proper nonconstant subalgebras.
P. J I P S E N E T AL.
412
ALGEBRA UNIV.
We now define a sequence of structures (U~, R~ ) that are essentially 7/copies
of the n-recession frame combined like a 7/-recession frame (see Figure 4(ii)). Let
U,, = {aij : i ~ 7/, 1 <_j <_n} and define
R~ = {(aij, a~,t) :i > k or (i = k a n d j > / ) }
w {(ai, l a ~ + l j ) : i e Z , 1 <_j<_n}
w {(aij, a~j+l):i~7/, l<_j<_n}.
For n = 1 the structure is (isomorphic to) the Y-recession frame, and it is easy to
check that, for fixed n and i, the set V~ = {a~j : 1 <_j _< n} together with the relation
R,, c~ (V i x V~) forms an n-recession frame whose complex algebra is generated by
{a;,~ }. The algebras ~3, are defined as ~ m f ( U , , R,, ).
THEOREM
Var(T0).
3. For each n ~ co the varieties Var(~3n) are distinct and cover
Proof We first show that ~3,, has no nonconstant subalgebras. Note that an
element of B, is either a finite subset of L~ or a union of a finite subset together with
U~= i ~ for some k ~ co or a complement of either such sets (since this collection of
sets is a subalgebra of ~ m ( U n , R, )). Let x E B, and suppose 0 < x < 1. Replacing
x by its complement if necessary, we may assume that f ( x ) < 1. F r o m the definition
of R~ it follows that f~(x) < 1, and it is easy to see that f3(x), f~(x) is one of the
I/~ for some k. C o n s e q u e n t l y f 3+ i(x), f z + ~(x) = Ilk + i f or each i ~ co. To generate the
set Vk _ ~, we choose the smallest m such that Vk _ i -~ gm(Vk), define y = gm(V1.), then
fZ(y) , f ( y ) = Vk_e. Finally gZ(V~).g(V~)={ai_2,~),
and now we can use the
operations f and g relativized to V)_2 (i.e. f ( x ) = f ( x ) 9 Vi_2 and ~(x) = g(x) 9 Vi_2)
to generate each singleton subset of Vi_ 2. So we conclude that ~B~ is generated by
each element x different from ~ or U~.
This however does not yet imply that Var(~3~) covers Var(To). Let 9.1 be a
subdirectly irreducible member of Var(~,,) that is not isomorphic to To. We need
to show that Var(~3,) ___Var(9.I). Since tense algebras are congruence distributive
we have ~I ~ H~P~,(~3~), and since B~ is a discriminator algebra (x r
implies
f ( x ) + g(x) = 1), we conclude that all members of SPr:(~B,) are simple and that (an
isomorphic copy of) 91 is among them. So we may assume that ~21 is a subatgebra
of ~ / F for some index set I and some nonprincipal ultrafilter F over L Consider
an element x ~ A such that 0 < x < 1. Again replacing x by its complement if
necessary, we assume thatf(x) < 1. Let J = {j ~ I : f ( x j ) < 1}. Then J e F a n d f ( x j )
generates all of Bn (in the j t h coordinate) uniformly for each j ~ J. But this implies
that f ( x ) generates a subalgebra of 9.1 isomorphic to ~ , , hence Var(~3,) ~ Var(91).
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
413
Since total tense algebras are discriminator algebras, the varieties V a r ( ~ n ) will
be distinct if we find a universal sentence ~bn that holds in ~ , and fails in ~m for
m < n. Let a =f3(x).f2(x), b =f4(x)-f3(x) and c = g(b).a = g(b). Then the sentence
~ n : O < f i x ) < l ~ a=gn(c)
[]
has the required property.
4. Application to symmetric semiassociative relation algebras
We n o w show that tense algebras are term-equivalent to reducts o f certain
semiassociative relation algebras. This allows us to extend the previous results to
varieties of semiassociative relation algebras.
A nonassociative relation algebra is of the f o r m 9.1 = ~A, + , , ;, ~, 1'), where
(A, + , - ) is a Boolean algebra, ~ is a unary operation, 1' is an identity element
with respect to the binary operation; and for all x, y, z e A
x.y;z=O
iff y . x ; ~ = O
iff
z.y;x=O.
(8)
( T h r o u g h o u t we use the convention that ; has precedence over - which in turn takes
precedence over + ) . The above equivalence can be expressed by equations, so the
class of all nonassociative relation algebras is a variety ( M a d d u x [10], Corollary
1.5, or [ 12], T h e o r e m 2). A nonassociative relation algebra is symmetric if it satisfies
the equation 2 = x. Every nonassociative relation algebra satisfies ( x ; y ) ~ = y ; 2
( M a d d u x [10], T h e o r e m 1.13(13)), so every symmetric nonassociative relation
algebra is commutative, that is, the operation ; is commutative. A nonassociative
relation algebra is reflexive if it satisfies x < x ; x.
I f we omit the requirement o f an identity element for a symmetric nonassociative relation algebra, then we obtain reducts (A, + , -, ;) which we will refer to as
symmetric r-algebras (symmetric residuated algebras). Thus (a, + , - , ;) is a symmetric r-algebra just in case (A, + , - ) is a Boolean algebra and for all x, y, z e A.
x.y;z=O
iff
y.x;z=O
iff
z.y;x=O.
(9)
Symmetric r-algebras are also commutative. Listed below are some other properties
o f nonassociative relation algebras and symmetric r-algebras that we will use later.
The first two equations assert that ; is normal, the next two that ; distributes over
414
P. J I P S E N E T A L .
ALGEBRA
UNIV.
+ , and the last two properties assert that ; is m o n o t o n e in each variable.
x;0=0;x=0
x;(y+z)=x;y+x;z
(x + y ) ;z = x ;z + y ; z
i f x < y then x ; z < y ; z
ifx<ythenz;x_<z;y.
The following two results establish a term-equivalence between the variety of
reflexive tense algebras and a subvariety of symmetric r-algebras.
T H E O R E M 4. Let 9.1 = ( A , + , - , f , g ) be a tense algebra and define a binary
operation ; on A by
x ; y = f ( x . y) + x . g(y) + y . g(x).
Then ( A , +, , ;) is a symmetric r-algebra that satisfies the equation x ; (x . y) <
x + y . Ifg, l is reflexive then ~A, +, -, ;) is reflexive, and the o p e r a t i o n s f a n d g can
be recovered f r o m ; by the term functions f ( x ) = x ; x and g(x) = x + x ; ft.
z.y.
P r o o f N o t e that z . x ; y = O
if
g(x) = 0, which is equivalent to
x.y.g(z)=O
and
z.x.g(y)=0
and
and
only
if
z.f(x.y)+z.x.g(y)+
z.y.g(x)=O.
This last statement is unchanged by permuting the variables x, y, and z, so the
statements z. x ; y = 0, x . z ; y = 0 and y 9 x ; z = 0 are equivalent. To check that
(A, + , -, ;) satisfies the equation x ; (`2 - y) _< x + y , we c o m p u t e
x ; (2.y) =f(x.
,2.y) + x . g ( 2 . y )
+`2.y.g(x)
< O+x +x.y
=x +y.
Assume that N is reflexive. Then x <_f(x) and x < g(x), so
x ; x = f ( x " x) + x g(x) + x g(x) = f ( x ) § x" g(x) = f ( x ) + x = f ( x ) ,
x + x ; "2 = x + f ( x . x) + x g("2) + "2 "g(x)
= x +f(0)
+ "2. g ( x )
= x . g ( x ) + "2 . g ( x )
= g(x),
and ~A, + , -, ;) is reflexive since x < f ( x ) = x ; x.
[]
Vol. 34, t995
Total tense algebras and symmetric semiassociative relation algebras
415
Since the equation x;(f.y)<<_ x + y is satisfied by all symmetric r-algebras
obtained in this way, we can only expect a converse for such algebras. A symmetric
r-algebra that satisfies this equation is called subadditive because for disjoint x and
y the relative product x ; y is below x + y. (Indeed, from x 9y = 0 and subadditivity
we get x ; y = x ; ( ~ - y ) + x ; ( x . y ) = x ;(37.y) + x ; 0 = x ; (5~,y) _<x + y . )
T H E O R E M 5. Let 9.1 = (A, +, -, ;) be a subadditive symmetric r-algebra and
define
f(x)=x+x;x
and
g(x)=x+x;2.
Then (A, + , - , f , g ) is a reflexive tense algebra. I f ~ is reflexive then ; can be
recovered from f and g by x ; y = f ( x . y) + x . g(y) + y . g(x).
Proof. We show that y .f(x) = 0 iff x . g ( y ) = O. It follows that (A, + , - , f , g )
is a tense algebra. Furthermore, (A, + , - , f , g ) is reflexive by the definitions o f f
and g. Suppose y -f(x) = y 9(x + x ; x) = O. Then y - x = 0 and y . x ; x = O, hence,
by distributivity, monotonicity, and subadditivity,
y . x ;.f = y - x
;(x-p
+ 2.p) =y.
x
;(x.y)
+ y . x ; ( 2 . y)
<_y" x ;x + y . ( x +37) = 0 .
This shows that x -y ; )7 = 0 by (9) and therefore x- g(y) = x . (y + y ; )7) = 0. For
the converse, suppose x . g ( y ) = 0 .
Then x - y = 0
and x . y ; f = 0 .
The
latter equation implies x ;y = y ; x <_y by commutativity and (9). It follows
that x - x ; y _ < x - y = 0 ,
hence also y . x ; x = 0 .
Consequently, y . f ( x ) =
y 9(x + x ; x) = 0. So far we know that (A, + , - , f , g ) is a reflexive tense algebra.
Next we show that
f(x
(lO)
. y) + x. g(y) + y . g(x) = x ; y + x . y.
First we expand the left side of (10).
f ( x .y) + x . g ( y ) + y .g(x) = x .y + ( x .y) ; ( x - y ) + x , ( y + y ;2)
+y.(x+x;2)
= x . y + ( x - y ) ;(x .y) + x . y
;)5 + y . x ;2.
Next, by distributivity, monotonicity, and subadditivity,
x.y ;p =x.y
;(p.x) +x.y
; ( y . 2 ) _<y ; x + x - ( y + 2 ) = x ; y + x . y ,
(11)
416
P. JIPSEN ET AL.
ALGEBRA UNIV.
and, similarly,
y- x ;2 _<x ;y + x . y ,
SO
x.y;2+y,x;2
<x;y+x.y.
Combining the previous equation with (11)
(x- y) ; (x- y) _< x ; y, we get half of (10), namely,
and
the
observation
that
f ( x . y) + x . g ( y ) + y . g(x) <_x ; y + x . y.
For the inclusion in the other direction, we first expand x ;y.
x ;y =(x.y +x.y) ;(x.y +2.y)
= (x. y) ; (x. y) ~2 (x.)7) ; (x. y) + (x. y) ; ()~. y) + (x. fi) ; ()~. y).
Next we show the last three terms in the expansion of x ; y
x- y ; 2 + Y" x ; 2. From monotonicity we have
(x. 2) ; (x. y) _<2 ; y
and by subadditivity,
(x2) ;(x-y) _<x.y +x.y
=x
SO
(x "2) ; (x. y) _<x 2 ;y.
In a similar way we also get
(~-y) ; (~. y) ~ y . x ;~.
By monotonicity,
(x-)9) ; ( i f . y ) _<x;)?.37 ; y
and, by subadditivity,
(x 2 ) ;(~ y ) _<x-2 + - ~ y _<x + y
are included in
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
417
SO
(x-y) ; ( ~ - y ) <_(x + y ) . x ; x . y ; y
=x.x
;x. y ;y + y.x
;Yc..f ;y
<_x. ~ ; y + y . x ;92.
We may now conclude that
x;y~(x.y);(x-y)+x-yLg+y.x;:2.
From this last equation and (11) we get the second half of (10), so (10) holds.
Now if 91 is reflexive, then x - y _< (x. y) ; (x- y) _< x ; y, hence x ; y + x - y =
x ;y, so by (10),
f ( x . y) + x . g ( y ) + y . g ( x ) = x ; y.
as desired.
[]
C O R O L L A R Y 6. The variety o f reflexive tense algebras is term-equivalent to the
variety o f reflexive subadditive s y m m e t r i c r-algebras.
Proof
Let 9.I = (A, + , - , f , g ) be a reflexive tense algebra. Define ; by
x ; y = f ( x . y) + x . g ( y ) + y - g(x).
By Theorem 4, (A, + , -, ;) be a reflexive subadditive symmetric r-algebra in which
the operations f, g are definable by f ( x ) = x ; x and g(x) = x + x ; Yc.
For the converse, let ~I = (A, + , - , ;) be a reflexive subadditive symmetric
r-algebra. Define f and g by
f(x) = x ; x
and
g ( x ) = x + x ; 92.
Reflexivity implies f ( x ) = x + x ; x , so by Theorem 5, (A, + , - , f , g3 is a reflexive
tense algebra in which ; can be recovered from f and g by x ; y = f ( x . y ) +
x- g(y) + y- g(x).
D
To establish a connection between subadditive symmetric r-algebras and total
tense algebras we require the semiassociative identity
(x;l);l=x;1
introduced in Maddux [9].
418
P. JI[PSEN ET AL.
A L G E BRA UNIV.
T H E O R E M 7. The variety generated by all total tense algebras is term-equivalent to the variety of reflexive subadditive semiassociative symmetric r-algebras.
Proof. Let ~A, + , - , J ~ g > be an algebra in the variety generated by all total
tense algebras, so that ~A, + , - , f , g ) is a tense algebra which satisfies the identity
x + f ( f ( x ) + g(x)) + g ( f ( x ) + g(x)) <_f(x) + g(x).
(12)
Let h(x) = f ( x ) +g(x) for every x e A. Then (12) is equivalent to
x + h(h(x)) _< h(x).
(13)
A consequence of (t3) is that ~A, + , - , f , g ) satisfies x <_h(x). From the latter
identity we can prove that ~A, + , - , f , g ) is reflexive. To see this, let z = x .f(x).
Then
z = x .f(x) < h(x .f(x)) = f ( x -f(x)) + g(x .f(x)) < f ( x ) + ff = 2.
From z _< ~ it follows that z = 0, hence x <_f(x). Define the binary operation ; on
A by
x ; y = f ( x . y) + x . g ( y ) + y . g(x).
(14)
By Theorem 4, (A, + , - , ;) is a reflexive subadditive symmetric r-algebra with
f ( x ) = x ; x and g ( x ) = x + x ; ~ . We will show (A, + , - , ;) also satisfies the
semiassociative identity. From (14) and the definition of h we have
x ; 1 = f ( x . 1) + x . g ( 1 ) + 1 .g(x) =h(x) + x - g ( t ) .
Since (A, + , - , f g ) is a tense algebra, the operations f and g are monotone and
distribute over +. It follows that the same properties apply to h, so we can derive
one half of the semiassociative identity from a consequence of (13), namely
h(h(x)) <_h(x), as follows:
(x ; 1) ; 1 = h(h(x) + x . g ( 1 ) ) + (h(x) + x . g(1)) . g ( 1 )
= h(h(x)) + h(x" g(1)) + h(x)" g(1) + x . g(l)
< h(x) + x . g(1)
=x;1.
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
419
For the other direction, note first that x < h(x) by (13), hence x <<_h(x) + x. g(1) =
x ; 1 for every x in A. Replacing x by x ; 1 yields x ; 1 < (x ; 1) ; 1.
Now suppose that {A, + , - , ;) is a reflexive subadditive semiassociative symmetric r-algebra. Define f and g by
f(x) = x ; x
and
g ( x ) = x + x ; x.
Reflexivity implies f ( x ) = x + x ; x, so (A, + , -, f, g ) is a reflexive tense algebra
and x ; y = f ( x . y ) + x . g ( y ) + y . g ( x ) by Theorem 5. We use the semiassociative
identity to derive (13). First note that h(x) = f ( x ) + g ( x ) = x ; x + x + x ; 2 =
x+x;(x+Y)=x+x;1.
Then
x + h(h(x))
= x +h(x
+ x ; 1)
=X +(x+x;1)+(x+x;
zX +x;
1);1
1 + ( x ; 1); 1
=x+x;1
= h(x).
Since (A, + , - , f , g ) is a reflexive tense algebra satisfying (13), which is equivalent
to (12), it is in the variety generated by all total tense algebras.
[]
Before we extend our results about total tense algebras to subadditive symmetric
semiassociative relation algebras, we recall what is known about the bottom of the
lattice ASA of semiassociative relation algebra varieties. J6nsson and Tarski [8]
proved that there are exactly three atoms, generated by the algebra 211,212 and 213,
in the lattaice ARA of varieties of relation algebras. Later Andrhka, J6nsson, and
N6meti [ 1] pointed out that the proof given there requires only the semiassociative
identity, so the same result holds in AsA. Furthermore, Var(211) has no join-irreducible covers and Var(212) has exactly one join-irreducible cover, generated by the
relation algebra of all binary relations on a 2-element set.
T H E O R E M 8. L e t 9.1 = ( A , +, -, ;, ~, 1') be a f i n i t e simple subadditive s y m m e t ric semiassociative relation algebra with m o r e than 4 atoms. Then 21 has a proper
n o n c o n s t a n t subalgebra.
A subalgebra of 21 is "proper", of course, if it is not the same as 21, but
the meaning of "nonconstant" is slightly different in this case, because semiassociative relation algebras have a distinguished identity element 1'. Indeed, every simple
Proof
420
P. J I P S E N E T AL.
ALGEBRA
UNIV.
finite semiassociative relation algebra has a subalgebra whose only elements are 0,
1, 1', and 1', and it is this subalgebra that we refer to as the "constant subalgebra"
of 91.
A nonassociative relation algebra is said to be integral if x ; y r 0 implies either
x r 0 or y ~ 0. If 9.1 is not integral, then 91 has a proper nonconstant subalgebra by
M a d d u x [11], T h e o r e m 3. We m a y therefore assume that 91 is integral. By M a d d u x
[11], T h e o r e m 4, this is equivalent to assuming that 1' is an atom.
Next we show that we m a y also assume 91 is reflexive. Let 9 1 ' =
(A, + , -, ;', ~, 1') where, for all x, y ~ A',
x;'y=x;y+x.y.
We claim that 91' is a reflexive finite integral symmetric semiassociatve relation
algebra with exactly the same subalgebras as 91. Obviously 91' satisfies 2 = x since
91 does so. Hence to prove that (8) holds for 91' we need only show (9) holds for
9.1', namely,
x . y ;'z =O
iff y . x ;'z =O
iff
z . y ; ' x =O.
Suppose x . y ; ' z = O .
Then O = x . ( y ; z + y . z ) = x . y ; z + x . y . z ,
so 0 =
x . y ; z and 0 = x 9y 9z. Using (8) and the s y m m e t r y of 91, we get 0 = y . x ; z, so
0=y'x;z+x'y'z=y'(x;z+x'z)=yx;'z.
The other equivalences can be
proved similarly. Next,
x;'l'=x;l'§
l'=x§
l'=x,
and, similarly, 1' ;' x = x, so 1' is an identity for ;'. To show that semiassociative
identity holds for 91', use the assumption that it holds for 91 to get
(x;'l)
;'1 = ( x ;
l+x.
1);'l=(x;1
+x);l+(x;l+x)-
1
=(x;1);l+x;l+x=x;l+x=x;'l.
So far we have shown that 9.1' is a symmetric semiassociative relation algebra.
Obviously 9.I' is finite. 91 and 91' have the same Boolean algebra as reduct, so 1' is
an a t o m in both, hence 91' is integral. 91' is reflexive since x < _ x ; x + x =
x ; x + x 9 x = x ;' x. W h a t remains is to show 91 and 91' have the same subalgebras.
The operations of 91' are term-definable from those of 91, so every subalgebra of 91
is a subalgebra of 91'. F o r the converse, assume ~ is a subalgebra of 91'. We wish
to show the universe B of ~ is closed under the operations of 91. The only
Vol. 34, 1995
Total tense algebras and symmetric semiassociative relation algebras
421
operation o f 92 which is n o t also an operation o f 92' is ;. Hence we need only show
that B is closed under ;. Let x, y E B. First expand x ; y as follows.
x ;y = (x .y) ; (x. y) + ( x - y ) ; (x. y) + (x -y) ; (2. y)
+ (x. y) ; (~ .y).
N o t e that ; and ;' coincide on disjoint arguments, that is, if u . v = 0, then
u;'v=u;v+u.v=u;v+0=u;v.
We can apply this observation to the last
three terms in the expansion o f x ; y, obtaining
x ; y = (x. y) ; (x .y) + (x. y) ;' (x- y) + (x. y) ;' (~. y) + (x 2 ) ;' (x "y).
Thus we need only show ( x . y ) ; ( x - y ) s
B. Let z = x ' y . W e claim that either
z<z;z,
in which case z ; z = z ; ' z s B ,
or else z - z ; z = 0 ,
in which case
z ; z = z ;' z . 5 e B. Suppose b o t h cases fail. Then u r 0 ~ v, where u = z 9 z ; z and
v = z.z;z.
Since 9.1 is integral and subadditive, we have 0 ~ u ; v < u + v. But
u ; v < < _ z ; z and u < z ; z ,
so O # u ; v < _ ( u - z ; z + v ) . z ; z = v . z ; z < v .
Thus
0 r u ; v- v. By (8) and sDnmetry, this yields 0 r v ; v- u. But v _< z and u _< z ; z, so
v ; v 9 u _< z ; z 9z ; z = 0, a contradiction. This completes the p r o o f o f our claim
that 92' is a reflexive finite integral symmetric semiassociative relation algebra and
9.I' has the same subalgebras as 92, so we might as well assume that 92 is reflexive.
Let 9 2 ' = ( A ' , + , - ' , ;') be the algebra whose universe A ' is defined by
A ' = {x ~ A : x < 1'}, where, for all x, y ~ A',
x-'=2.
x,
1',
y=x;y
We claim that 92' is a subadditive reflexive symmetric r-algebra. First o f all,
(A, + , - ' ) is a Boolean algebra. Since 9.1 is symmetric, (9) holds. W h e n (9) is
applied to 9.1', that is, with ; replaced by ;', it is equivalent to
x-y;z=l'=O
iff y . x ; z .
1'=0
m
iff
_
_
z.y;x-
i'=0.
_
_
But for x, y, z ~ A ' we have x = x 9 1', y = y 9 1', and z = z 9 1', so the latter condition
actually coincides with (9). Thus 92' is a symmetric r-algebra. The subadditivity and
reflexivity o f 92' follows immediately f r o m the subadditivity and reflexivity o f 92.
By T h e o r e m 5, 92' is term-eqtfivalent to a reflexive tense algebra 9 2 " =
(A',+
-,f,g), wheref(x)=x+x,
x=x+x
x . 1' and g ( x ) = x + x ;
x =
422
P. J I P S E N ET AL.
ALGEBRA UNIV.
m
x+x;(21'). 1'. N o t e that i f 0 r
thenx<l'
andx;l=l
since 9.1 was
assumed to be integral, so f ( x ) + g ( x ) = x + x ; ( x .
1--). l ' + x ; ( 2 .
1'). 1 ' =
x + x ; 1'- l ' = x ; l ' l'+x;l',
l'=x;1.
1 ' = 1'. But 1' is the unit element o f
9.I', so 9.1" is total. 9.I' has more than 3 atoms, so we can apply T h e o r e m 1 to the
a t o m structure of ~I" and conclude that 9,1" has a proper n o n c o n s t a n t subalgebra.
O f course the same conclusion holds for 91', since 91' is term-equivalent to 96". Let
B' be the universe of this subalgebra of 9.1', and define B = B' t2 {x + 1' ~ A 9 x ~ B}.
Then, for all x, y e B', we have
x ; y = x ; ' y + x ; y . 1',
(x+l');y=x;'y+y+x;y.
l',
(x + 1') ; ( y + 1') = x ; ' y + x + y + 1'.
These equations, together with the assumption that 1' is an atom, show that
(B, + , -, ;, 1') is a proper nonconstant subalgebra of ~1.
[]
Since the preceding theorem did not assume reflexivity, there are more than four
finitely generated varieties covering Var(gX3). The exact n u m b e r is f o u n d counting
the n u m b e r o f nonisomorphic structures that can be obtained by deleting loops
(x, x ) from the a t o m structures o f T1, T2, T3 and T 4. Nonfinitely generated covers
o f Var(9.13) can be constructed from the total tense algebras ~3n o f T h e o r e m 3 via
the term-equivalence, similar to the p r o o f above.
C O R O L L A R Y 9. In the lattice o f varieties o f subadditive symmetric semiassociatire relation algebras there are exactly 19 finitely generated varieties, and infinitely
many nonfinitely generated varieties, that cover Var(~13).
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Department of Mathematics
Iowa State University
Ames, IA 50011-2066
U.S.A.