Convex Semilattices in Algebraic Effects

• Convex semilattices are algebraic structures that integrate a commutative idempotent semilattice join with barycentric convex-combination operations for probabilistic choice.
• They model the monad of non-empty finitely generated convex sets of finitely supported probability distributions, capturing both nondeterministic and probabilistic effects.
• Key results include the unique base theorem, canonical free convex semilattice construction, and representation in Riesz spaces, which underpin normalization and cancellation properties.

Searching arXiv for recent and foundational papers on convex semilattices and related structures. Convex semilattices are algebraic structures that combine a semilattice operation for nondeterministic choice with barycentric operations for probabilistic choice. In the standard formulation, a convex semilattice is a set equipped with a commutative idempotent semilattice join and a family of binary convex-combination operations indexed by parameters in $(0,1)$, subject to barycentric axioms and a distributivity law linking the two kinds of choice. This structure presents the monad of non-empty, finitely generated convex sets of finitely supported probability distributions, so it functions simultaneously as an equational theory and as a semantic model for combined probabilistic and nondeterministic effects (Bonchi et al., 2020).

1. Algebraic definition

The standard signature consists of a binary operation $\vee$ and binary operations $+_p$ for each $p \in (0,1)$. The operation $\vee$ satisfies the semilattice axioms

$$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$

while the operations $+_p$ satisfy the barycentric axioms

$x +_p x = x,\qquad x +_p y = y +_{1-p} x,$

and

$$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$

The boundary cases behave as expected: $x +_1 y = x,\qquad x +_0 y = y.$ The interaction axiom is

$\vee$0

Using symmetry of $\vee$1 and commutativity of $\vee$2, the symmetric rule

$\vee$3

is derivable and is used in normalization arguments (Bonchi et al., 2020).

This axiomatization isolates the intended operational reading. The semilattice join models nondeterministic choice, whereas the barycentric operations model probabilistic choice. The distributivity law states that mixing after a nondeterministic choice coincides with taking the nondeterministic choice among the corresponding mixtures. In later work the semilattice operation is often written $\vee$4 rather than $\vee$5, but the axioms are the same (Sokolova et al., 15 Jul 2025).

A frequent source of confusion is that the phrase “convex semilattice” does not always denote exactly this signature in the literature. In the present sense, it refers to a structure carrying both a genuine semilattice operation and a full barycentric algebra. Distinct but related usages occur in ordered-algebraic convexity and in older convex-space literature; those variants are addressed below.

2. Convex sets of distributions and the presentation theorem

Let $\vee$6 be a set, and let $\vee$7 denote the set of finitely supported probability distributions on $\vee$8, i.e. functions $\vee$9 with finite support and $+_p$0. For $+_p$1 and $+_p$2, their convex combination is defined pointwise by

$+_p$3

More generally, if $+_p$4 and $+_p$5 with $+_p$6, then $+_p$7 is the distribution $+_p$8 with

$+_p$9

For $p \in (0,1)$0, its convex hull is

$p \in (0,1)$1

A convex set is finitely generated if it is the convex hull of finitely many distributions. Writing $p \in (0,1)$2 for the collection of all non-empty, finitely generated convex subsets of $p \in (0,1)$3, one obtains the endofunctor

$p \in (0,1)$4

On $p \in (0,1)$5, the semilattice and convex operations are

$p \in (0,1)$6

and

$p \in (0,1)$7

The latter is a Minkowski convex combination, and it satisfies the barycentric axioms pointwise. The unit is

$p \in (0,1)$8

and the multiplication $p \in (0,1)$9 is given by

$\vee$0

The monad laws hold, and $\vee$1 is a homomorphism of convex semilattices: $\vee$2 This yields the categorical statement that the algebraic theory of convex semilattices presents the monad $\vee$3: the free convex semilattice on $\vee$4 is $\vee$5, and the Eilenberg–Moore category $\vee$6 is equivalent to the category of convex semilattices (Bonchi et al., 2020).

The scope of this presentation is precise. It concerns non-empty, finitely generated convex sets of finitely supported distributions. The restriction to finite generation and finite support ensures the compactness and finiteness properties used in the proofs and in the monad structure. The syntax may be restricted to rational parameters for convenience, but the axioms and semantics are valid for real $\vee$7 (Bonchi et al., 2020).

3. Bases, extreme points, and canonical forms

For $\vee$8, a base is a finite set $\vee$9 such that

$$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$0

and for each $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$1,

$$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$2

Equivalently, a base is a minimal generating set with no redundant element. The central structural theorem is that every $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$3 has a unique base (Bonchi et al., 2020).

One proof is functional-analytic. Viewing $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$4 with the product topology, finitely generated convex subsets are compact convex sets, so Krein–Milman yields

$$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$5

where $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$6 is the set of extreme points of $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$7. Minimality then forces $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$8 to be the smallest generating set, hence the unique base. A second proof is combinatorial: if two bases generate the same convex set, each base element can be expressed as a convex combination of the other base, and the resulting coefficient matrices force a bijection between the two index sets, so the bases coincide up to permutation (Bonchi et al., 2020).

The theorem has an algebraic role beyond convex geometry. It provides canonical generators for elements of the monad $$x \vee x = x,\qquad x \vee y = y \vee x,\qquad (x \vee y)\vee z = x \vee (y \vee z),$$9 and underpins normal forms in the equational theory. When a term is rewritten into a nondeterministic–probabilistic form, redundant probabilistic summands can be removed exactly when their denotations lie in the convex hull of the others. The unique base theorem guarantees that this elimination is canonical, which is the key step in constructing the inverse of the presentation map and proving that the presentation is an isomorphism (Bonchi et al., 2020).

The finite-generation hypothesis is essential to the stated result. The paper proves uniqueness for finitely generated convex sets of finitely supported distributions and notes that in infinite settings uniqueness of bases need not hold without further compactness or extremal assumptions (Bonchi et al., 2020).

4. Free convex semilattices and basic examples

The free convex semilattice on a set $+_p$0 is

$+_p$1

Its elements are exactly the non-empty, finitely generated convex sets of finitely supported distributions over $+_p$2, and the unit sends $+_p$3 to the singleton $+_p$4. Its universal property is the expected one: for any convex semilattice $+_p$5 and any function $+_p$6, there exists a unique homomorphism $+_p$7 extending $+_p$8, obtained by interpreting convex hulls as joins and convex combinations as the operations $+_p$9 (Bonchi et al., 2020).

A standard two-point example takes $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$0 with Dirac distributions $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$1 and $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$2, and

$x +_p x = x,\qquad x +_p y = y +_{1-p} x,$3

Then

$x +_p x = x,\qquad x +_p y = y +_{1-p} x,$4

has unique base $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$5. If $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$6, then

$x +_p x = x,\qquad x +_p y = y +_{1-p} x,$7

which is the entire $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$8-simplex on $x +_p x = x,\qquad x +_p y = y +_{1-p} x,$9. For the Minkowski convex combination,

$$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$0

contains

$$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$1

These examples show concretely how the semilattice and barycentric operations interact at the level of convex sets of distributions (Bonchi et al., 2020).

This construction is also the semantic basis for the monadic interpretation. The join is not set-theoretic union but convex union, and the probabilistic operation is not pointwise addition of sets but the set of all pairwise mixtures between the two arguments. That distinction is essential: it is exactly what makes the free algebra closed under both nondeterministic and probabilistic composition.

5. Cancellativity and representation in Riesz spaces

A convex semilattice is cancellative if its underlying convex algebra is cancellative, meaning that for any $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$2,

$$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$3

The relevant classical result for convex algebras is the Stone–Kneser theorem: a convex algebra is cancellative if and only if it is isomorphic to a convex subset of a real vector space with the canonical affine operations. Recent work extends this to convex semilattices (Sokolova et al., 15 Jul 2025).

The extension states that a convex semilattice $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$4 is cancellative if and only if it is isomorphic to a convex subset $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$5 of a Riesz space $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$6 that is closed under binary suprema, with operations

$$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$7

where $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$8 is the lattice join in $$(x +_q y) +_p z = x +_{pq}\Bigl(y +_{\frac{p(1-q)}{1-pq}} z\Bigr).$$9. Thus cancellativity forces a representation in a lattice-ordered vector space rather than merely in an affine space (Sokolova et al., 15 Jul 2025).

The proof proceeds in stages. First, Stone–Kneser embeds the convex-algebra part into a real vector space. Second, the paper introduces the perspective shift

$x +_1 y = x,\qquad x +_0 y = y.$0

and proves that for $x +_1 y = x,\qquad x +_0 y = y.$1 and $x +_1 y = x,\qquad x +_0 y = y.$2, the restriction $x +_1 y = x,\qquad x +_0 y = y.$3 is a convex semilattice homomorphism. Third, it constructs a linear subspace

$x +_1 y = x,\qquad x +_0 y = y.$4

and a well-defined extension $x +_1 y = x,\qquad x +_0 y = y.$5 of the semilattice operation to $x +_1 y = x,\qquad x +_0 y = y.$6. Finally, the order induced by $x +_1 y = x,\qquad x +_0 y = y.$7,

$x +_1 y = x,\qquad x +_0 y = y.$8

turns $x +_1 y = x,\qquad x +_0 y = y.$9 into a Riesz space, with infima given by

$\vee$00

No extra semilattice axioms are required beyond the convex semilattice laws; cancellativity of the convex part alone suffices for the representation theorem (Sokolova et al., 15 Jul 2025).

Canonical examples are supplied by familiar vector lattices. The coordinatewise order on $\vee$01 yields lattice join by coordinatewise maximum, so any convex subset closed under coordinatewise suprema becomes a convex semilattice. Function spaces $\vee$02 with pointwise order provide another class of examples. These models realize probabilistic choice by affine combination and nondeterministic choice by lattice join in the ambient ordered linear structure (Sokolova et al., 15 Jul 2025).

6. Variants, extensions, and related notions

A broader semiring-based formulation replaces barycentric operations by semimodule structure. For a semiring $\vee$03 satisfying the conditions needed for a canonical weak distributive law of the powerset monad over the $\vee$04-semimodule monad, the resulting composed monad sends a set $\vee$05 to the convex subsets of the free $\vee$06-semimodule $\vee$07. Its Eilenberg–Moore algebras are structures carrying both left $\vee$08-semimodule operations and complete semilattice operations, subject to distributivity. In the finitary restriction, the algebras are semilattices with bottom together with left $\vee$09-semimodules satisfying interaction axioms such as $\vee$10, $\vee$11 for $\vee$12, and $\vee$13. This construction differs from the non-empty theory principally by including the empty convex set, which gives a bottom element and yields a CPPO-enriched Kleisli category (Bonchi et al., 2020).

An enriched generalization appears in the theory of sober $\vee$14-convex spaces and $\vee$15-join-semilattices. There the truth values form a complete residuated lattice $\vee$16, convexity is carried by an $\vee$17-convex structure on $\vee$18-subsets, and the semilattice side is provided by suprema of nonempty finite $\vee$19-subsets in an $\vee$20-ordered set. The Scott $\vee$21-convex structure satisfies the compatibility law

$\vee$22

and sobrification yields a canonical $\vee$23-join-semilattice completion. The main equivalence states that an $\vee$24-ordered set $\vee$25 is an $\vee$26-join-semilattice completion of $\vee$27 if and only if the Scott $\vee$28-convex space $\vee$29 is a sobrification of $\vee$30 (Wu et al., 2024).

A separate line of work uses “convex semilattice” in a more combinatorial sense. In the theory of convex spaces, a semilattice can be viewed as a convex space of combinatorial type by setting $\vee$31 to be constant on $\vee$32; for a meet-semilattice, one puts

$\vee$33

together with the boundary laws $\vee$34 and $\vee$35. These structures satisfy the abstract convex-space axioms but do not embed into vector spaces as convex subsets. This is related to convex semilattices in the algebraic-effects sense, but it is not the same signature, because the semilattice operation there is not an additional operation interacting with an independent barycentric algebra; it is the entire nontrivial mixture law (0903.5522).

Ordered-algebraic convexity gives another distinct usage. In a join-semilattice regarded as an ordered structure, convex subsets may be defined to be subsemilattices, the convex hull of a set is the subsemilattice it generates, and the extreme points of a convex subset are its coirreducible elements. In compact locally convex topological semilattices one then has a Krein–Milman-type theorem: $\vee$36 with

$\vee$37

This notion concerns convexity internal to semilattice order rather than the probabilistic–nondeterministic algebra described above (Poncet, 2013).

Convex geometries supply yet another semilattice-centered context. For an algebraic convex geometry, the compact closed sets form a join-semilattice $\vee$38, and the global order-scatteredness of the lattice of closed sets is governed by semilattice obstructions. In finite join-dimension, the key criterion is that $\vee$39 is order-scattered iff $\vee$40 is order-scattered and $\vee$41 is not embeddable into $\vee$42 as a join-subsemilattice. In the special case of relatively convex sets, the corresponding obstruction is $\vee$43 (Adaricheva et al., 2015).

Taken together, these developments show that “convex semilattice” names a stable core idea—the interaction of convexity and idempotent join—but the exact formalization depends on whether the setting is probabilistic algebra, semimodule theory, enriched order, abstract convex spaces, or ordered semilattice convexity. The standard algebraic-effects formulation remains the one in which convex semilattices present the monad of non-empty finitely generated convex sets of finitely supported probability distributions and serve as the canonical Eilenberg–Moore algebras for combined nondeterministic and probabilistic choice (Bonchi et al., 2020).