Keimel’s problem on the algebraic axiomatization of convexity

Convex sets may be viewed as algebras equipped with a set of binary convex combinations that is indexed by the open unit interval of real numbers. Convex sets generate the variety of barycentric algebras, which also includes semilattices where the semilattice multiplication is repeated uncountably many times. Barycentric algebras are defined by three axioms: idempotence, skew-commutativity, and skew-associativity. Since the skew-associativity axiom is rather complicated, Klaus Keimel has asked whether it can simply be replaced by the entropic law. It turns out that the answer is negative. The counterexamples presented and studied in this paper are known as threshold barycentric algebras, depending on a threshold taken from the left-hand side of the closed unit interval. They offer an entire spectrum of algebras, ranging from barycentric algebras for threshold 0 to commutative idempotent entropic groupoids for threshold 1/2.


Introduction
Real convex sets may be presented algebraically as sets with binary operations given by weighted means, the weights being taken from the open unit interval I • =]0, 1[ in the set of reals. The class of convex sets is a quasivariety (defined by certain implications), and generates the variety (defined by identities) of socalled barycentric algebras (Section 2). Both these classes have a well-developed theory, a special case of the general theory of modes (idempotent and entropic algebras, Section 3) [3,4,11,12,13,14,15,16,18]. In particular, like all modes, barycentric algebras implement self-distributivity in the sense of [2].
The variety of barycentric algebras contains convex sets, semilattices (where all the barycentric operations coincide), and certain semilattice sums of convex sets. It is defined by three types of axioms: idempotence, skewcommutativity, and skew-associativity, the latter two implying the entropic property that each operation is a homomorphism. Since the skew-associativity axiom is rather complicated, Klaus Keimel asked whether it may simply be replaced by entropicity alone [8]. In this paper, we show that the answer is negative. We construct an entropic algebra, of the same type as barycentric algebras, which is idempotent and skew-commutative, but is not skew-associative (Theorem 5.1).
Together with barycentric algebras, the counterexample belongs to a much larger family of algebras described as threshold barycentric algebras, which model the concept of threshold convexity introduced in Section 4. Take  Varieties of threshold barycentric algebras fall into three classes: • The variety B 0 of threshold-0 barycentric algebras coincides with the variety B of usual (skew-associative) barycentric algebras. • For 0 < t < 1/2, threshold-t barycentric algebras are not skew-associative (Theorem 6.4). But the variety B t of threshold-t barycentric algebras is equivalent to the variety B of extended barycentric algebras (Theorem 9.4). Extended barycentric algebras are defined like usual barycentric algebras, but with the closed unit interval I of barycentric operations replacing the open interval I • , and thus including projection dimonoid structure. • The variety B 1/2 of threshold-(1/2) barycentric algebras is equivalent to the variety CBM of extended commutative binary modes (Theorem 7.2). These algebras have one binary commutative mode operation that is given by t = 1/2, together with projection dimonoid structures.
Thus threshold barycentric algebras offer an entire spectrum of algebras, ranging from the usual barycentric algebras at one end to the commutative binary modes at the other. In [17], it was shown that barycentric algebras may be used to replace Boolean networks in systems biology with models offering a more realistic tracking of the biochemistry. Because the Boolean networks correspond to the use of threshold-(1/2) convexity in that application, the threshold t may be viewed as a parameter providing a smooth transition between the two classes of models. Theorem 8.5 shows that for a threshold 0 < t < 1/2, finite-dimensional simplices (free barycentric algebras) are also generated by their vertices under the basic threshold-t barycentric operations. The golden section makes an appearance in the analysis underlying the proof of the theorem (Remark 8.3).
Section 10 discusses the varieties S t of threshold-t semilattices (Proposition 10.3). Extended semilattices are defined by adding projection operations to semilattices, in similar fashion to extended barycentric algebras and extended commutative binary modes. Then for each positive threshold t, the variety S t of threshold-t semilattices is equivalent to the variety of extended semilattices (Proposition 10.6).
Sections 11 and 12 investigate the behavior of threshold-t barycentric varieties within the full variety SC of skew-commutative modes of barycentric algebra type I • × {2}. Theorem 12.3 identifies the meet B s ∧ B t for distinct thresholds s and t, while Theorem 12.6 identifies the join B s ∨ B t .
Background facts concerning barycentric algebras and commutative binary modes are summarized in Sections 2 and 3. For more information on such algebras, and modes in general, we refer the reader to the monographs [14,16]. We use notation and conventions similar to those of the cited monographs and [21].

Barycentric algebras
Let F be a field. A unary operation of complementation is defined by for p, q ∈ F . A binary operation of implication is defined by of skew-associativity for all p, q ∈ I • . The class B of barycentric algebras forms a variety [3,4,11,12,13,14,15,16,18]. Convex subsets of real affine spaces are barycentric algebras (C, I • ) under the operations xyp = xp + yp = x(1 − p) + yp (2.4) for each p ∈ I • . They form the subquasivariety C of the variety B defined by the cancellation laws (xyp = xzp) → (y = z) (2.5) for all operations p of I • . In particular, cancellative members of B are precisely convex sets (see [16,Th. 5.8.6]), and the class C generates the variety B. Other examples are provided by semilattices, idempotent, commutative semigroups (S, ·), considered as barycentric algebras with the "stammered" operation x · y = xyp for all p ∈ I • . Note that semilattice barycentric algebras (S, I • ) satisfy the identities xyp = xyr for all p, r ∈ I • . They are also interpreted as ordered sets, meet semilattices with the ordering relation defined by x ≤ y if and only if x · y = x. The variety S of semilattice barycentric algebras is the only non-trivial proper subvariety of the variety B.
The structure of general barycentric algebras (A, I • ) is described by the following theorem. Barycentric algebras may also be defined as algebras (A, I), where I = [0, 1] is the closed unit interval of R, with the operations 0 and 1 defined by xy0 = x and xy1 = y (2.6) [3,4,12,13,18]. The class B of barycentric algebras defined in this way is also a variety. It is specified by the identities defining B together with the additional identities (2.6). Examples are provided by convex sets and semilattices considered as usual barycentric algebras, with two additional operations 0 and 1. We will refer to members of B as extended barycentric algebras.

Entropic algebras and modes
For an algebra (A, Ω) equipped with a set Ω of binary operations, and ω, ϕ ∈ Ω, consider the entropic identity (xyω)(ztω)ϕ = (xzϕ)(ytϕ)ω. Further examples of modes are provided by barycentric and extended barycentric algebras, and by affine spaces and their reducts. (Here, affine spaces over a field F of odd characteristic are defined as algebras (A, F ) with binary operations p given by (2.4) for each p ∈ F . In a similar way, one may define affine spaces over commutative rings of odd characteristic [16]). Among groupoids (or magmas), algebras with one binary operation, we will be especially interested in commutative binary modes, groupoid modes (A, ) with a commutative multiplication , and in extended commutative binary modes

Threshold convexity
Convexity is described algebraically using the binary operations (2.4) of weighted means, for complementary weights p , p taken from the open unit interval. The new concept of threshold convexity replaces weighted means whose weights differ widely by projections to the most heavily weighted argument. (a) For elements x, y of a convex set C, define Proof. For elements u, v, w, x of C and 0 < r, s < 1, the entropic identity must be verified. Note that the identity is symmetrical in the operations r and s.
There are various cases for the evaluation of the two sides of (4.3), according to whether the threshold-convex combinations r and s are small, moderate, or large. If both are small, the algebra (C, r, s) is a left zero semigroup (taken with repeated operations). As such, it is entropic. Dually, (C, r, s) is entropic as a right zero semigroup if both r and s are large. Again, (C, r, s) is entropic as a reduct of a barycentric algebra if both r and s are moderate.
If one combination (say r) is small, and the other is moderate, then both sides of (4.3) reduce to uw s. The case where one is large, and the other moderate, is dual. Finally, suppose r is small and s is large. Then the lefthand side of (4.3) evaluates to uw s = w, while the right-hand side evaluates to wx r = w.  and skew-commutative. By Proposition 4.4, I is entropic. But it will now be shown that I is not skew-associative: the identity (2.3) written as

Threshold barycentric algebras
Definition 6.1. Set a threshold 0 ≤ t ≤ 1/2. Then the class B t of threshold -t (barycentric) algebras is the variety generated by the class of convex sets under the threshold-convex combinations of Definition 4.1.
Following Definition 4.1, the operations r of threshold-t barycentric algebras will be called extreme (small if 0 < r < t, and large if t < r < 1), and If t > 0, then a threshold-t barycentric algebra C t = (C, I • ) defined on a convex set C may be considered as an algebra (C, t ] is the set of operations r = r for t ≤ r ≤ t . In particular, if t = 1/2, then C 1/2 = (C, 1/2 = 1/2, {r | r < 1/2}, {r | r > 1/2}). Remark 6.3. Threshold-(1/2) convex sets are commutative binary modes under the operation = 1/2. As such, they generate the variety CBM [16,Ch 5]. Note that the elements 0 and 1 of the algebra (R, ) generate the unit dyadic interval D 1 .
In view of Remarks 6.2 and 6.3, it emerges that threshold barycentric algebras offer a spectrum reaching across from barycentric algebras at one end through to commutative binary modes at the other.
The proof of Theorem 5.1 involved an algebra in the class B 1/2 which is not skew-associative. The following result uses a slightly more refined and general adaptation of that construction. Proof. As noted in Remark 6.2, algebras in the class B 0 are barycentric algebras. As such, they are certainly skew-associative. Now suppose that the threshold t is positive. Consider the closed unit interval I = [0, 1] as a convex set. In the skew-associativity identity (5.1), set p = t and q = 1 − t. Then (6.1) Take x = 1 and y = z = 0. Using the inequality (6.1) and the definition (4.1), along with the idempotence of the threshold-convex combinations, the right-hand side of (5.1) evaluates to However, the left-hand side evaluates to (10 t) As we have already seen, cancellativity plays an important role in the theory of barycentric algebras and commutative binary modes. The concept is extended to threshold barycentric algebras as follows.
Definition 6.5. Consider a threshold 0 ≤ t ≤ 1/2, and a threshold-t barycentric algebra (C, I • ). The algebra is cancellative if it satisfies the quasi-identity Remark 6.6. If a threshold-t barycentric algebra were to satisfy the cancellation quasi-identity (6.2) for any extreme combination r, then it would necessarily be trivial.
The final result of this follows directly by the known properties of barycentric algebras and commutative binary modes (compare Sections 2 and 3.)
Lemma 7.1 extends to an equivalence between the corresponding varieties. Proof. First recall that the variety CBM of commutative binary modes is generated by the 1/2-subreducts of dyadic affine spaces, and hence also by 1/2-reducts of real convex sets.
Lemma 7.1 shows that the -reducts of convex sets, considered as extended commutative binary modes, are equivalent to threshold-(1/2) barycentric algebras. Thus each member of CBM becomes a threshold-(1/2) barycentric algebra. On the other hand, each threshold-(1/2) barycentric algebra has the structure of an extended commutative binary mode under the operation = 1/2, with ¡ equal to any small extreme operation and £ equal to any large extreme operation.
The lattice of varieties of commutative binary modes was first described in [6]. (See also [14, §4.5] and [16, §10.4].) The lattice of proper subvarieties of CBM is isomorphic to the product of the two-element lattice and the lattice of odd natural numbers under the divisibility relation. Each proper subvariety of CBM is defined by one binary identity. The addition of extreme operations r for r ∈ ]0, 1/2[ ∪ ]1/2, 1[ to CBM-algebras has no influence on the lattice of subvarieties of these algebras. Thus the lattice of subvarieties of B 1/2 is isomorphic to the lattice of subvarieties of CBM. Each proper subvariety of B 1/2 is defined by one binary identity defining the corresponding subvariety of CBM.

Simplices
Recall that, under the operations of I • , the real k-dimensional simplex is the free algebra in B over the finite set X = {x 0 , . . . , x k } of its vertices. It may be characterized as the I • -subreduct of the free real affine space R k over the same set X. In particular, the closed unit interval (I, I • ) is the free barycentric algebra on two free generators 0 and 1. The aim of this section is to present Theorem 8.5, showing that for a threshold 0 < t < 1 2 , each real simplex is generated by threshold-convex combinations of its extreme points.
First recall a fact we will use in the following proofs. For each p ∈ I • , and subalgebras A, B of a convex set (C, I • ), the set ABp := {abp | a ∈ A, b ∈ B} is a subalgebra of (C, I • ). See [16, §5.1], for example. Proof. For a positive integer h, one has

Induction on h then shows that each interval [t h , (t ) h ] lies in A.
For a positive integer k, one has Thus with K as in (8.1), the union forms the connected set ]0, (t ) K ], and the result follows.
shows that x may be generated from {0, 1}. By symmetry, each element of ] 1 2 , 1 − L 2 [ may also be generated from {0, 1}. Since it follows that the entire interval I is generated by {0, 1}.

The variety B t with 0 < t < 1/2
The aim of this section is to characterize the variety B t of threshold-t algebras for 0 < t < 1/2. First note that the extreme operations of threshold-t algebras have no influence on the generation of the algebras considered in Proposition 8.4 and Theorem 8.5. This has an important consequence. Note the following consequence of Theorem 8.5. The following theorem may be proved in similar fashion to Theorem 7.2.

Theorem 9.4. Set a threshold 0 < t < 1/2. Then each variety B t of thresholdt barycentric algebras is equivalent to the variety B of extended barycentric algebras.
As another consequence of Proposition 8.4 and Theorem 8.5, one obtains the following. with moderate operations r = r for t ≤ r ≤ t and the original extreme operations, is the variety B t of threshold-t barycentric algebras.
The skew-associativity that appears in Theorem 9.5(d) concerns the derived binary operations, but not the basic extreme binary operations. In principle, one could write out the corresponding identities explicitly, with repeated use of the moderate basic operations, but the expressions involved would be very complicated.

Threshold-t semilattices
Recall that each barycentric algebra has its semilattice replica as a homomorphic image. Moreover the following theorem holds.
As each of the varieties B t , for 0 < t < 1/2, may be considered as the variety of barycentric algebras with additional extreme operations, it is easy to see that each member of such a variety also has a semilattice replica, however equipped additionally with extreme operations of the type of B t -algebras. This leads to the following definition. (a) Equality between the respective moderate operations p, for t ≤ p ≤ t ; (b) Associativity for each moderate operation p, for t ≤ p ≤ t .
Proof. For 0 < t < 1/2, idempotence of the moderate operations follows directly by Theorem 9.5(a). Then again by Theorem 9.5(a), one has skewcommutativity for the moderate operations. Given equality of the moderate operations, and the symmetry of the interval [t, t ], their commutativity follows.
For the case t = 1/2, where 1/2 is the only moderate operation, the identities of Proposition 10.3(a) vanish, while Proposition 10.3(b) just specifies the associative law for the commutative binary mode operation 1/2 . By Remark 4.2, the case t = 0 reduces to the usual situation for extended barycentric algebras, where the equalities (a) suffice, and the associativity (b) then follows from the skew-associativity. Remark 10.4. If t = 0, then the variety S 0 is the variety S of (stammered) semilattices. If t = 0, then the variety S t is equivalent to the variety of (stammered) semilattices equipped with additional extreme operations r for r < t and r > t .
As a direct consequence of Theorem 10.1, one obtains the following. Recall (Section 7) that the variety B 1/2 contains infinitely many subvarieties.
Corollary 10.5. Consider a threshold 0 ≤ t < 1/2. Then the variety B t of threshold-t barycentric algebras contains only one proper non-trivial subvariety, namely the variety S t of threshold-t semilattices.
Extended semilattices may be defined in similar fashion to extended barycentric algebras and extended commutative binary modes. In particular, we have the following analogue of Theorems 7.2 and 9.4. Proposition 10.6. For each positive threshold t, the variety S t of threshold-t semilattices is equivalent to the variety S of extended semilattices.

Barycentric words and identities
For 0 ≤ t ≤ 1/2, let SC t be the variety of algebras of type I • × {2}, defined by the identities of idempotence, skew-commutativity (2.2) and entropicity (3.1) for all operations of I • , along with the left-zero identities for all operations p with 0 < p < t and right-zero identities for all operations p with t < p < 1.
In particular, for t = 0, there are no extreme operations, and SC 0 coincides with the variety SC of skew-commutative modes of type I • ×{2}. On the other hand, if t = 1/2, then all the operations p for p = 1/2 are extreme. Note that the varieties SC t form a chain, where for 0 = t < u, one has SC u ≤ SC t , and SC t < SC 0 = SC for all t = 0. Each variety B t is a subvariety of SC t . For t = 1/2, the varieties B 1/2 and SC 1/2 coincide.
Note also that idempotence and entropicity alone do not define the variety SC. For example, consider an I • -algebra (A, I • ), where for each p ∈ I • , one has xy p = x. While the algebra A is idempotent and entropic, it is not skewcommutative.
Proposition 11.1. For each 0 ≤ t < 1/2, the variety B t is a proper subvariety of the variety SC t .
Proof. First, note that the variety B = B 0 is a proper subvariety of SC. Now assume that t = 0. It suffices to provide an example of an algebra which belongs to SC t , but not to B t . Let A be the algebra (I, I • ) defined on the real unit interval I with I • -operations defined as follows: Vol. 79 (2018) for 0 < p < 1 and t < u < 1/2. Simple calculations show that the algebra A satisfies the identities defining the variety SC t , but it is not a threshold-t barycentric algebra.
We will now describe free SC t -algebras for t = 0. Consider a countable set X = {x i | 0 < i ∈ Z} of variables. Let T denote the set of finite, binary rooted trees. For an element T ∈ T , let T l denote the set of leaves of T . Let T n denote the set of nodes of T , including the root. Then let λ T or λ : T l → X and ν T or ν : T n → I • be functions. Each such function pair (λ T , ν T ), for each element T ∈ T , specifies a word in the language of SC-algebras recursively as follows.
Definition 11.2. Let T be a binary rooted tree.
(a) If T has a unique vertex, then T l is a singleton and T n is empty. If the unique value of λ T is x i , then that variable is the word specified by the function pair (λ T , ν T ). A tree T ∈ T is complete if each non-leaf node has two children, and all leaves have the same level. The word specified by a complete tree is also called complete. Note that for a complete tree of height h, the number of leaves is equal to l = 2 h , and the number of nodes is n = 2 h+1 − 1.
Let W be the set of words specified by the trees of T . In a variety SC t , with threshold 0 < t ≤ 1/2, such words may be reduced to certain special words. First note that by skew-commutativity, for any words w 1 and w 2 and any large operation p, the SC t -algebras satisfy w 1 w 2 p = w 2 w 1 p , where p is small. This allows us to change all appearances of large operations in w into small ones. Next, observe that each small operation p satisfies xy p = x, and more generally w 1 w 2 p = w 1 . This allows one to replace each appearance of subwords of the form w 1 w 2 p in w by w 1 . In this way, one obtains a word w t without symbols of extreme operations, and with all operation symbols belonging to the set [t, t ]. Then, using the idempotent laws for all operations of [t, t ], one may extend the word w t to a complete word w t c as follows. If w t contains a subword of the form u = xv p, where v is already a complete word specified by a tree T with the function pair (λ T , ν T ), replace x with the word v 1 specified by the tree T with the function pair (λ T , ν T ) such that ν T = ν T , and with λ T assigning the variable x to each leaf of T . The word w t c obtained Proof. The proof is by induction on the number n of operational symbols p from the set [t, t ]. If n = 1, then the words have the form x 1 x 2 p, where p ∈ [t, t ] and x 1 and x 2 may be equal or different. Hence they are in canonical form. A longer element not in canonical form may be expressed as w 1 w 2 p, where w 1 and w 2 are canonical words, and p ∈ I • . If p is a small operation, then w 1 w 2 p = w 1 , while if p is a large operation, then w 1 w 2 p = w 2 , and hence both are in canonical form. Finally, if p is a moderate operation, then w 1 w 2 p is in canonical form.

Varieties of threshold algebras
Recall that for a threshold t, the symbol S t denotes the variety of threshold-t semilattices.
Proposition 12.1. For any two thresholds s and t with 0 ≤ s < t ≤ 1/2, the varieties S s and S t are incomparable. Moreover, the meet S s ∧S t is the variety T of trivial algebras.
Proof. First note that if 0 ≤ s < t ≤ 1/2, then the set of extreme operation symbols of S t -algebras contains the set of extreme operation symbols of S salgebras. Hence the set of identities defining extreme operations of S t contains the set of identities defining extreme operations of S s . To show that S s and S t are incomparable, consider an operation r for s < r < t. Then the identity xy r = x holds in S t , but is not satisfied in S s . On the other hand, the variety S s satisfies the commutative law xy r = yx r, which is not satisfied in S t . The same laws may be used to show that the meet S s ∧ S t is trivial. Indeed, the Note that for t = 0, each variety B t and its subvariety S t are strongly irregular. By the theory of regularized varieties (see e.g. [16, §4.3]), it follows that for t = 0, the regularization B t of any variety B t , and similarly the regularization S t of any variety S t , consists precisely of P lonka sums of algebras in the original variety, and is defined by the regular identities true in that variety. The regularization is the join of the original variety and the variety S of semilattices. Note also that in the regularization, the left-zero operations become left-normal operations, and the right-zero operations become rightnormal operations [16, §4.3]).
In what follows, we will be interested in joins of varieties B t .
Definition 12.5. Set thresholds 0 ≤ s < t ≤ 1/2. Set B s,t to be the variety of idempotent, entropic, skew-commutative I • -algebras defined by the following identities: (1) xy p = x for all p < s ; (2) xy p = y for all p > s ; (3) all identities true in the variety B t mod of Definition 9.1.
The algebra (I, I • ) with appropriately defined extremal operations may be a member of any variety B t . As a member of B t for t = 0, it will be denoted by I t . Proof. First recall that for any 0 < r < 1/2, each B r -algebra satisfies the identity xy p = x for all p < r, and all identities true in B q mod for q ≥ r. Since s < t, it follows by Definition 12.5 that any identity true in B s,t is satisfied in both the varieties B s and B t , and hence in B s ∨B t . Consequently, B s ∨B t ≤ B s,t .
To verify the converse inequality, we will show that each identity true in both B s and B t (and hence in B s ∨ B t ) is also satisfied in B s,t . First note that all left-zero and all right-zero identities true in B s also hold in B s ∨ B t , and in B t,s . Now let w = v (12.1) be an identity satisfied in B s ∨B t containing operation symbols p for s ≤ p ≤ s . Using skew-commutativity, we may reduce this identity to an identity without large extreme operations. So in what follows, we assume that all the symbols of extreme operations appearing in (12.1) are symbols of left-zero operations.
If all the operation symbols appearing in (12.1) belong to [t, t ], then the identity is satisfied by all B t -algebras, and hence by all B t mod -algebras. Consequently, it holds in all B s,t -algebras. It follows that if the identity (12.1) contains only symbols of left-zero operations p for all p < s, and/or operations from the set [t, t ], then it holds in all B s,t -algebras.
In what follows, assume that (12.1) contains operation symbols p only in the range s ≤ p ≤ s . First assume that one side of (12.1), say w, contains only operation symbols p for s ≤ p < t, and the other contains some p with t ≤ p ≤ t . Consider the B t -algebra I t . Substitute 0 and 1 for the variables of (12.1) in any way. Then in I t , w(0, 1) = 0, while v(0, 1) > 0, contradicting the fact that (12.1) holds in B t . Thus if one side contains only p with s ≤ p < t, then the same holds also for the other side.
So now assume that both sides of (12.1) contain some operations from [t, t ], or all operation symbols of (12.1) are from [s, t]. By Proposition 9.3, each operation p for s ≤ p < t may be written as a composition of operations q for t < q < t . Hence the identity may be written as an identity true in all B s,t -algebras. Corollary 12.7. Let 0 ≤ s, t, u, w ≤ 1/2 be thresholds, and let s < t and u < w. Then the variety B s,t is a subvariety of B u,w if and only if u ≤ s < t ≤ w.
Proof. First assume that u ≤ s < t ≤ w. Note that each B s,t -algebra A satisfies all identities true in B t mod , and moreover xy p = x for all p < u. Hence A is a member of B u,w , and consequently B s,t ≤ B u,w . Now let B s,t ≤ B u,w . Assume on the contrary that u > s or t > w. First let u > s. Note that the B s -algebra I s is a member of B s,t , and hence of B u,w . In particular, each B s -algebra has to satisfy the identity xy p = x for each p ∈]s, u[. However, in the algebra I s , we have 01 p = 01 p = 0, a contradiction. Now assume that t > w. Then each B t -algebra, and in particular the algebra I t , must be a member of B u,w and has to satisfy each identity true in