Barycentric algebras and beyond

Barycentric algebras are fundamental for modeling convex sets, semilattices, affine spaces and related structures. This paper is part of a series examining the concept of a barycentric algebra in detail. In preceding work, threshold barycentric algebras were introduced as part of an analysis of the axiomatization of convexity. In the current paper, the concept of a threshold barycentric algebra is extended to threshold affine spaces. To within equivalence, these algebras comprise barycentric algebras, commutative idempotent entropic magmas, and affine spaces, all defined over a subfield of the field of real numbers. Many properties of threshold barycentric algebras extend to threshold affine spaces.


Introduction
This paper is part of a series that is devoted to extensions of the concept of barycentric algebras. Barycentric algebras form a variety generated by convex sets, when the latter are viewed as algebras equipped with the set of binary convex combinations indexed by the open real unit interval I • , subject to the hyperidentities of idempotence, skew-commutativity, and skew-associativity.
The first extension examined barycentric algebras over subfields of R [25, §5.8]. Next, certain subrings of R were brought into consideration [2,3,4], in particular the ring Z[1/2] of dyadic numbers [12,13,14,15], and other principal ideal subdomains of R [22]. In [18], it was observed that when the closed unit real interval was used to index the basic operations of a barycentric algebra, then the interval naturally assumed the structure of an LΠ-algebra, one of the such algebras are proved. The final section describes meets and joins of varieties of threshold affine spaces over the field F . In particular, Theorem 6.4 provides a correction to the comparable results of [9, §12], where one case was omitted.
From this paper and [9], it is clear that all (symmetrical) intervals of F containing 1/2 may be used to define affine spaces or barycentric algebras. However, they do not all play an equally important role in defining such algebras. The most fundamental intervals are the unit interval and the whole line F , which both carry the structure of a dual monoid with involution in the sense of [8]. Investigations of further extensions will concentrate on finding the appropriate detailed algebraic structure of such intervals, and then using this structure as a source of basic operations for further extension of the barycentric algebra concept.
Background facts concerning convex sets, barycentric algebras and affine spaces are summarized in Section 2. Readers may also consult the references at the end of this paper, and a newer survey provided in [21]. For additional information on such algebras, and modes in general, see the monographs [23,25]. Notation and conventions generally follow those of the cited monographs and [27].

Modes, affine spaces and barycentric algebras
In the sense of [23,25], modes are defined as algebras where each element forms a singleton subalgebra, and where each operation is a homomorphism. For algebras (A, Ω) of a given type τ : Ω → N, these two properties are equivalent to satisfaction of the identity of entropicity for each pair ψ, φ of operators in Ω. One of the main families of examples of modes is given by affine spaces over a commutative unital ring R (affine R-spaces), or, more generally, by subreducts (subalgebras of reducts) of affine spaces. Here, affine spaces are considered as Mal'tsev modes, as explained in the monographs [23,25]. In particular, if 2 is invertible in R, an affine R-space may be considered as the reduct (A, R) of an R-module (A, +, R), where R is the family of binary operations for each r ∈ R. The class of all affine R-spaces is a variety [1], denoted by R. If 2 is invertible in R, then the variety R is defined by the idempotent and entropic laws, together with the trivial laws xy0 = x, xy1 = y (2.3) and the affine laws [xyp] [xyq] r = xy pqr (2.4) for all p, q, r ∈ R. An important class of subreducts of affine spaces is given by convex sets, defined as subreducts of affine R-spaces. Convex sets are characterized as subsets of a real affine space closed under the operations r of weighted means taken from the open real unit interval I • = ]0, 1[. Thus a convex set contains, along with any two of its points, the line segment joining them. The class C of convex sets, considered as algebras (C, I • ), generates the variety B of barycentric algebras, and forms a subquasivariety of B [17].
The definitions of convex sets and barycentric algebras may be readily extended to the case of a subfield F of R, with its own unit interval I • = {s ∈ F | 0 < s < 1} [25,Chapters 5,7]. For p, q ∈ I • , set p := 1 − p, and define the dual product p • q := (p q ) = p + q − pq. Then the variety B of barycentric algebras over F is defined by the identities of idempotence for each p in I • , the identities of skew-commutativity for each p in I • , and the identities of skew-associativity for each p, q in I • [25,Section 5.8]. It is worthy of note that idempotence, skew-commutativity, and skew-associativity may all be construed as hyperidentities of algebras (A, I • ) (in the sense of [16]). As reducts of affine spaces, barycentric algebras are entropic. They comprise convex sets, so-called stammered semilattices where p = q for all p, q ∈ I • , and certain sums of convex sets over (stammered) semilattices.
Barycentric algebras may also be axiomatized as extended barycentric algebras (A, I), where I is the closed unit interval {s ∈ F | 0 ≤ s ≤ 1} of F , with the operations 0 and 1 defined by as respective left and right projections. Note that skew-associativity in the form (2.7) is then no longer a hyperidentity of (A, I), since 0 • 0 = 0, and q/(p • q) is not defined for p = q = 0. The class B of extended barycentric algebras is a variety, specified by the identities (2.5)-(2.7) defining B together with the additional identities (2.8). For more information about barycentric algebras, see [5,7,18,19,22,24,26,28,29].

Affine spaces and barycentric operations
We now extend the concept of a barycentric algebra by enlarging the set of basic operations, (possibly) weakening the algebraic structure of that set, while retaining as many key properties of barycentric algebras as possible. Since skew-associativity is essential to the definition of barycentric algebras [9], we will discuss this identity in the more general setting of affine spaces. We first introduce some operations defined on any field F , recalling that division is not an operation in the sense of universal algebra. For p, q ∈ F , a binary operation of implication is defined by When F is the two-element field GF(2) = {0, 1}, the implication (3.1) becomes the usual Boolean implication, while the dual product p • q becomes the usual Boolean disjunction. Consider the operation of (dual ) division on the field F . Note that otherwise.
Another operation of interest is Note that otherwise.
If F is a subfield of R, and I • is the open unit interval of F , then for p, q ∈ I • one has q < p • q, whence q/(p • q) ∈ I • and the skew-associativity may be written as xy p z q = x yz p • q → q p • q.
(3.4) or xy p z q = x yz p q p • q.
(3.5) This form of skew-associativity will be called right skew-associativity. As shown in [18], the identity also holds for all p, q in the closed unit interval I = [0, 1] of F . In fact, it may be observed directly that the original skew-associativity (2.7) holds if at least one of p, q is 1, and if precisely one of p, q is 0. The only critical value of p and q is p = q = 0. The identities (2.5), (2.6), and (3.4), together with (2.8), provide an axiomatization for extended barycentric algebras.
Another form of skew-associativity for p, q ∈ I • is given by the identity which may also be written as using (3.3). This form of skew-associativity will be described as left skewassociativity. As in the case of right skew-associativity, it may be verified that left skew-associativity holds for all s, t in the closed interval I. While for right skew-associativity the only critical values of p, q were p = q = 0, in the present case they are p = q = 1.
Lemma 3.1. Let A be a nontrivial affine F -space over a subfield F of R, and let p, q ∈ F . (a) A satisfies the right skew-associativity Proof. If p • q = 0, then the proof of the right skew-associativity (3.5) proceeds as in the usual case p, q ∈ I • for barycentric algebras [25, §5.8]. If p = q = 0, then the proof proceeds as discussed above for extended barycentric algebras [18]. Conversely, suppose p • q = 0 and (p, q) = (0, 0). Note that q = 1 implies q = 0 and p • q = (p q ) = 0 = 1, so q = 1 under the current assumptions, and p = q/(q − 1). Since A is nontrivial, it contains distinct points a and b. Then Since q = 0, we have abq = a. It follows that the right skew-associativity does not hold in this case. The treatment of the left skew-associativity is dual.
The concluding observations of this section are concerned with other relations between axioms of barycentric algebras over F and affine F -spaces.

Binary reducts of affine F -spaces
In this section, we consider affine F -spaces over a fixed subfield F of R, with the open and closed unit intervals of F denoted respectively by I • and I. Recall that a convex subset C of an affine F -space A is just a subreduct of A with respect to the operations belonging to I • . We will replace the interval I • by an open interval ]q, q [, where q is any member of F not exceeding 1/2, and q = 1 − q. Then ]q, q [ will denote the set of operations {r | r ∈ ]q, q [}.
The subreducts of a given type of algebras in a given (quasi)variety form a quasivariety [11, §11]. In particular, the class C q of all q-convex subsets of affine F -spaces is a quasivariety.

Definition 4.2.
Let q ∈ F with q ≤ 1/2. Then the variety B q generated by the quasivariety C q is called the variety of q-barycentric algebras.
Note that C 0 is the quasivariety C of usual convex sets, and B 0 is the variety B of barycentric algebras. A special relationship between the quasivarieties C q and the varieties B q emerges from consequences of the following general facts.
Let R be a subring of the ring R. Recall that, under the operations of R, the affine R-space R k (for k ∈ N) is the free algebra, in the variety R of affine R-spaces, over a finite set X = {x 0 , . . . , x k } of free generators. The free algebra is [ 25, §6.3]. In particular, the line R is the free affine R-space on two free generators x 0 = 0 and x 1 = 1. Recall the following. Proposition 4.3 [20]. Let R be a commutative, unital ring. Let Ω be a set of affine combinations over R. Let ΩR be the quasivariety of Ω-subreducts of affine R-spaces. Let J be the free ΩR-algebra on two generators.
(a) The free ΩR-algebra XΩR over a set X is isomorphic to the Ω-subreduct, generated by X, of the free affine R-space XR.
Finite-dimensional simplices are free barycentric algebras. Furthermore, the quasivariety C q and the variety B q have the same free algebras [11, §13].
Corollary 4.4. The free B q -algebra over X is isomorphic to the ]q, q [-subreduct, generated by X, of the free affine F -space XF over X.
Proof. The proof is by induction on k. First note that 01t = t and 01t = t .
Note that t t < t < t < (t ) 2 , and more generally Proof. The inductive proof is similar to the proof of [9, Thm. 8.5], with simplices Δ n replaced by the affine F -spaces F n , and the extreme points of the simplices replaced by the free generators of the space F n . For n = 1, the theorem follows by Proposition 4.6. Now suppose that the result is true for a positive dimension n. Consider the affine F -space F n+1 with free generators x 0 , x 1 , . . . , x n+1 . It consists of all affine combinations of a subspace F n generated by n + 1 free generators of F n+1 , say by x 0 , x 1 , . . . , x n , and the generator x n+1 . Thus an arbitrary point x, which is not on a hyperplane Π n parallel to F n , lies on a line going through a point p of F n and the point x n+1 . By Proposition 4.6, the point x is generated under the operations of [t, t ] by p and the generator x n+1 . If a point x belongs to the hyperplane Π n , then it lies on a line going through a point y generated by some p of F n and x n+1 , and a point q of F n . As before, the line is generated by y and q. By induction, the points p and q are generated under the operations of [t, t ] by free generators of F n , which of course are also free generators of F n+1 . Thus x is generated under the operations of [t, t ] by the free generators of F n+1 .  Proof. The first two conditions follow by results of [9, §9]. If 0 ≤ q < 1/2, then the variety B q coincides with the variety B t mod of t-moderate barycentric algebras of [9, §9]

Threshold algebras
Threshold barycentric algebras were introduced by the authors in [9] as a tool to analyze the axiomatization of barycentric algebras. We first recall the definition. Take  Threshold barycentric algebras offer an entire spectrum of algebras, ranging from the usual barycentric algebras at one end (where t = 0) to the (extended) commutative binary modes at the other (for t = 1/2). Theorem 8.5 of [9] 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. This implies that such threshold-t barycentric algebras are equivalent to extended barycentric algebras.
In this section, we will extend the concepts of a threshold and threshold barycentric algebras to threshold affine spaces. (The same could be done for the q-barycentric algebras of Definition 4.2. But since they are equivalent to affine spaces anyway, it will suffice to consider affine spaces alone). First note that the definitions and results of [9] remain true in the case of subfields of the field R. As in the previous section, F will always denote a fixed subfield of R, and I • and I will denote the respective open and closed unit intervals of F . The following definition generalizes the concept of threshold convex set.
(a) For elements x, y of an affine F -space A, define for r ∈ F . Then the binary operations r are described as threshold -t affine combinations.
If 0 < t < 1/2, then the variety A t of threshold-t affine F -spaces is equivalent to the variety B t of threshold-t barycentric algebras, and hence to the variety B of extended barycentric algebras [9, §6]. The variety A 0 is also equivalent to the variety B of extended barycentric algebras.
These observations may be summarized in the following classification theorem. The regularization of a given variety is the variety that is defined by the regular identities holding in the given variety. The regularization of a regular variety is that same variety. Since the variety B = B 0 of barycentric algebras is regular, its regularization B coincides with B. On the other hand, if t = 0, then threshold-t barycentric algebras form a (strongly) irregular variety B t . Its regularization B t , consisting of P lonka sums of B t -algebras, does not coincide with B t .
Note that the variety A = F may also be defined as the variety of skewcommutative modes, of the type of F -algebras, satisfying the trivial identities (2.3), the affine identities (2.4), and the binary Mal'tsev identities Definition 5.7. For t ∈ F with t ≤ 1/2, let A t mod be the variety generated by the reducts A, [t, t ] of threshold-t affine F -spaces with respect to moderate threshold combinations. Members of A t mod are said to be t-moderate affine Fspaces.
The variety A 0 mod coincides with the variety B of extended barycentric algebras, and A 1/2 mod coincides with the variety CBM of commutative binary modes. By Theorem 4.3, the free A t mod -algebra over X is isomorphic to the [t, t ]-subreduct, generated by X, of the free affine F -space XF . Moreover if t > 0, then by [9, Thm. 8.1], the free A t mod -algebra is equivalent to the free barycentric algebra XB over X, and if t < 0, then by Theorem 4.7, it is equivalent to the free affine F -space XF over X. The following corollary extends Theorem 9.5 of [9]. Corollary 5.8. Let −∞ < t < 1/2. Then the variety A t of threshold-t affine F -spaces is defined by the following identities: (a) Idempotence, skew-commutativity and entropicity for all operations of F ; (b) The identity xy r = x for each small operation r; (c) The identity xy r = y for each large operation r; and (d) Skew-associativity for the (derived) binary operations that are generated by the moderate operations in the case when t > 0, and the trivial, affine and binary Mal'tsev identities for (derived) binary operations generated by the moderate operations in the case t < 0.
Remark 5.9. Both threshold barycentric algebras and threshold affine spaces fall into the following more general scheme, which may be interesting in its own right, even though we currently have no further examples. Consider a variety V of Ω-algebras such that Ω is the disjoint union Ω 1 ∪Ω 2 of Ω 1 and Ω 2 , and, by composition, the Ω 2 -operations generate the Ω 1operations. Let W be the variety generated by the Ω 2 -subreducts of V-algebras. We additionally assume that free algebras in both varieties generated by the same set of generators are the same. Then the varieties V and W are equivalent. Now replace each (n-ary) operation ω in Ω 1 by some trivial operation ω, where

Varieties of threshold algebras
Meets and joins of varieties A t of threshold affine F -spaces may be described using methods similar to those employed for threshold barycentric algebras in [9] (noting the correction of [9, Th. 12.6] originally provided in [10], and discussed below following the proof of Theorem 6.4).
Recall that for 0 ≤ t < 1/2, the varieties A t are equivalent to the variety B of extended barycentric algebras, and for −∞ < t < 0, they are equivalent to the variety A of affine F -spaces. If 0 < t < 1/2, then the variety A t contains as a unique non-trivial subvariety the variety S t of extended semilattices, where all operations from [t, t ] are equal and associative, and the extreme operations reduce to one left and one right projection. For t < 0, the varieties A t have We now consider joins of varieties A t . First, we will define the varieties SC t , in a similar way as in [9, §11], but this time for all elements t of the field F . This means that SC t is the variety, of the type of F -algebras, defined by the identities of idempotence, skew-commutativity and entropicity for all operations of F , along with left-zero identities for all operations p with p < t and right-zero identities for all operations p with t < p. Each variety A t is a subvariety of SC t . In a variety SC t , each word may be reduced, using the left-zero and right-zero identities and skew-commutativity, to its reduced form w t without extreme operations, and containing only moderate operations r belonging to [t, t ]. It follows that each identity w = v satisfied in A t may be written in its reduced form w t = v t , only containing symbols of moderate operations, and satisfied in the variety A t mod . Definition 6.2. Let s, t ∈ F ∪ {−∞}. Set thresholds −∞ ≤ s < t ≤ 1/2. Let A s,t be the variety of idempotent, entropic, skew-commutative algebras, of the same type as F -algebras, defined by the following identities: (1) xy p = x for all p < s ; (2) xy p = y for all p > s ; Proof. For r < 1/2, each A r -algebra satisfies the identities xy p = x for all small operations p, the identities xy p = y for all large operations p, and all identities true in A u mod , for u ≥ r, that only involve moderate operations. Since s < t, it follows by Definition 6.2 that any identity true in A s,t is satisfied in both the varieties A s and A t , and the same holds for the identities defining the  [t, t ], and then the identity reduces to the identity w t = v t true in A t mod . It follows that the identities true in both A s and A t satisfy the conditions of Definition 6.2. Hence they hold in A s,t , and A s,t ≤ A s ∨ A t .
Note that a small change in Definition 6.2 and Theorem 6.4 will provide a correction to Theorem 12.6 of [9], where one case was lost [10]. It is sufficient to assume that F = R and 0 ≤ s < t ≤ 1/2. Recall that in this case the varieties A t and B t are equivalent.
Note that the variety A s,t is a proper subvariety of the variety A s,t . This is shown by Example 6.5 below. First observe that the algebra (F, F ), with appropriately defined operations, may be considered as a member of each of the varieties A t and A s,t . As a member F t of A t , it satisfies the identities that define A t , and as a member F s,t of A s,t , it satisfies the identities that define A s,t . Example 6.5. Let −∞ < s < 1/5 and 2/5 < t < 1/2. Let p = 1/4 and q = 1/5. Then p • q = 2/5 and q/(p • q) = 1/2. Since s < p, q, p • q < t, it follows that the variety A s satisfies skew-associativity for p = 1/4 and q = 1/5. On the other hand, the same identity holds in A t , since in this case both of its sides are equal to x. It follows that the identity holds in A s,t . Now consider the algebra F s,t but satisfying additionally the following conditions: xy1/5 = y and xy2/5 = x, and moreover xy4/5 = x and xy3/5 = y.
It is easy to see that this algebra is a member of A s,t . However, it does not belong to A s,t . The left-hand side of skew-associativity for p = 1/4 and q = 1/5 equals z, whereas the right-hand side equals x.
The following example indicates some relations between the variety A = A −∞ = F of affine F -spaces and the variety A 0 equivalent to the variety B of extended barycentric algebras. Example 6.6. Note that the varieties A, A 0 , S 0 , the trivial variety T and the variety A −∞,0 form a lattice isomorphic to N 5 . The variety A −∞,0 satisfies all identities of A 0 mod , in particular all the identities defining the variety B of barycentric algebras (since affine spaces satisfy all of them), then all the identities true in affine F -spaces which in A 0 -algebras have both sides equal to the same variable. Among them are some (but not all) skew-associative identities, and some affine identities containing only operation symbols p for p < 0 or p > 1. No binary Mal'tsev identities are satisfied in A −∞,0 .