Weakly distributive homomorphism¶

A join-semilattice homomorphism that lifts every two-piece cover of an image to a matching two-piece cover in its source.

Version: v1 · 2026-10-25 · History
Domain-specific #: 14455
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Universal Algebra, Semilattice Theory → Mathematics

Core Idea¶

A weakly distributive homomorphism is a map between join-semilattices that preserves joins and can lift a split made in its target. Write the map as μ:S→T. For every c∈S and a,b∈T satisfying μ(c)≤a∨b, there must be x,y∈S with c≤x∨y, μ(x)≤a, and μ(y)≤b.^[1] The quantifier ranges over every such source element and target pair, not just a convenient split. The source and target of each variable matter: c,x,y belong to S; a,b belong to T.^[1]

Join preservation concerns what happens when source elements are combined and then mapped. The additional lifting law asks the converse style of question: when an image lies below two target pieces joined together, can a source cover be chosen whose images fit the corresponding pieces? A map can pass the first test and fail the second. This difference gives the named class its identity rather than treating weak distributivity as a synonym for homomorphism.

Structural Signature¶

Typed semilattice pair. S and T each have a join operation and its induced order.^[1] Without both structures, neither the target cover nor the source lift is defined.
Join-preserving map. μ:S→T is a homomorphism for the join operation. In the canonical compact-congruence examples it also preserves zero, but that extra property is not inserted into the general definition.^[1]
Universal target cover test. For every c∈S and a,b∈T, the premise μ(c)≤a∨b activates the test.^[1] Verifying only a favored pair does not certify the map.
Matched source lift. The witnesses x,y∈S satisfy c≤x∨y, μ(x)≤a, and μ(y)≤b.^[1] These three inequalities distinguish the class from generic join-homomorphisms.

What It Is Not¶

It is not the claim that the target semilattice itself obeys a distributive lattice law. The law here constrains a map and the way target covers lift to its source. It is also not the distinct notion of a distributive congruence on one semilattice. The frozen source article juxtaposes that property with weakly distributive maps; sharing a page does not make it a constitutive part of this map. Schmidt's earlier formulation and later monomial-congruence work have different definitions and qualifications, so their labels cannot substitute for the quantified map test.^[2]^[3]

Nor does join preservation alone suffice. Let S={0,1} be a two-element chain, let T={0,a,b,1} be a four-element Boolean join-semilattice, and set μ(0)=0, μ(1)=1. This map preserves zero and joins. Yet μ(1)=a∨b, while any two source elements whose join covers 1 include 1 in at least one position. Its image is target top, below neither atom. The required lift therefore fails. This is a direct finite deduction from the definition, not a counterexample taken from Wehrung's presentation.

Scope of Application¶

The type belongs to join-semilattice and universal-algebra arguments where a map between ordered join structures must preserve more than the join operation. Wehrung gives two theorem classes of such maps on compact congruences: a reduct-to-expansion map for algebras on the same carrier, and a convex-sublattice-to-enclosing-lattice map.^[1] Both use canonical maps that send a compact source congruence to the target congruence it generates.^[1] Their hypotheses matter; they do not establish that every map between compact-congruence semilattices has the lifting property.

The same definition can be tested for other proposed join maps, but a use outside these theorem classes requires its own proof for all eligible covers. A finite check is decisive for a particular finite pair only if every c,a,b triple is covered; a single failed triple disproves membership.

Clarity¶

The type separates three assertions that are easily conflated: a map preserves joins; a target cover of one image exists; and that cover has suitably bounded source witnesses. The first does not entail the third. Wehrung's definition places c,x,y in the source and a,b in the target.^[1] Stating that typing explicitly permits a proof or counterexample without guessing what the symbols mean.

Manages Complexity¶

Instead of tracing every individual congruence through a particular construction, one can ask whether the canonical map has the all-cover lifting property. A theorem establishing it for a construction then packages infinitely many cover tests under explicit hypotheses. That compression is safe only while the reduct relation or convexity assumption remains in force. The label should not hide which map is used, which structures provide joins, or where the theorem's hypotheses enter.

Abstract Reasoning¶

To assess a proposed example, first type S, T, and μ and verify join preservation. Then fix arbitrary c∈S and a,b∈T satisfying the cover premise. A positive proof constructs x,y meeting all three inequalities, possibly by applying an established theorem with matching hypotheses. A negative proof needs just one triple for which no pair works. The Boolean-target map above shows the latter method: a target split of top exists, but the source has no corresponding pieces with bounded images.

Knowledge Transfer¶

Within universal algebra, the definition transfers literally to different join-semilattice maps because its carriers, operation, quantifiers and lift have the same meaning. The reduct and convex-sublattice constructions show two different ways a canonical compact-congruence map can satisfy it. Their shared mathematical field should not be presented as cross-domain evidence. Outside join-semilattice settings, one may recognize the broader idea of a structure-preserving function through the parent Homomorphism and ultimately Function (Mapping), but the specific two-piece lift cannot be asserted without joins and an order in both carriers.

Examples¶

Reduct and expansion¶

Let A be a reduct of an algebra B on the same carrier, so B has the operations of A and possibly more. The source S=Con_c(A) and target T=Con_c(B) are join-semilattices of compact congruences. The canonical map sends an A-congruence to the B-congruence it generates. Wehrung states that this (∨,0) map is weakly distributive.^[1]

For a finite carrier illustration, take the empty-signature algebra on {0,1,2} as A, and add to B a unary operation with f(0)=f(1)=0, f(2)=1. Con(A) has all five partitions; Con(B) has three congruences: equality, the partition identifying 0 and 1, and the universal congruence. The generated-congruence map is not the identity. This finite calculation is author-derived; because the target congruences form a chain, it is a carrier illustration rather than a hard two-piece split test.

Mapped back: Con_c(A) and Con_c(B) fill the typed pair; generated congruence is the join-preserving map; arbitrary target congruence covers are the universal test; the reduct theorem supplies source congruences satisfying the matched lift. The finite calculation shows the carriers and map but does not replace that theorem.

Convex sublattice¶

Let K be a convex sublattice of a lattice L. The canonical map Con_c(K)→Con_c(L) extends each compact congruence by generating an L-congruence. Wehrung states that this (∨,0) map is weakly distributive. Convexity is a hypothesis, not an optional descriptive detail.^[1]

For a finite illustration, take the chain L={0<a<1} and its interval K={0,a}. Con(K) has two elements; Con(L) has four. The nonzero congruence of K maps to the congruence of L that collapses the lower adjacent pair. The target image is a join-prime atom, so this example again displays the typed carriers and canonical map more than it stresses a difficult split.

Mapped back: the two compact-congruence semilattices are the typed pair; generated congruence is the join-preserving map; covers of its images are the universal test; the convex-sublattice theorem warrants the matched source lift. Removing convexity takes the construction outside the cited theorem's guarantee.

Structural Tensions¶

The cited definition and theorem classes establish a condition and examples, but they do not establish an intrinsic tradeoff between competing objectives. No structural tension is asserted here merely to fill a template. The useful diagnostic distinction is proof obligation: preservation of joins is checked directly, whereas weak distributivity also requires every qualifying target split to lift.

Structural–Framed Character¶

The entry is strongly structural within mathematics: its truth depends on typed semilattices, a map and quantified inequalities, not on evaluation or institutional approval. Its vocabulary travels literally across semilattice and congruence constructions, but the named law does not automatically travel outside ordered join structures. Human practice affects how one discovers or proves examples, not the condition itself; the historical name arose within algebra, but that origin does not make the formal test culturally contingent. Cases can be recognized wherever the same map-and-lift structure exists rather than imported by analogy. Function (Mapping) carries the portable single-valued assignment; Homomorphism adds preservation of declared algebraic operations; this entry adds the semilattice-specific all-cover lift. Its character: structural and domain-specific, with a precise literal transfer boundary.

Structural Core vs. Domain Accent¶

The portable skeletal relation is a single-valued assignment from a specified source to a specified target; Function (Mapping) owns that relation. Homomorphism contributes the narrower algebraic rule that declared operations are preserved. This entry adds the domain-bound mechanism: join-induced order, every two-piece target cover, and bounded source witnesses. If those joins and inequalities are stripped away, the weakly distributive identity disappears even though a function remains. The named entry therefore does not clear the Prime bar through superficial analogies to splitting or reverse transport. Any broader reusable pattern would require independently attested instances in unlike domains and a separately admitted identity; it is not silently claimed here.

This entry is a kind of Homomorphism.

The strict parent is Homomorphism: every instance preserves joins as an algebraic operation, and the lifting law is its extra differentia. Homomorphism's the broader abstraction is now Function (Mapping), the general single-valued mapping structure; the former strict Transformation link has been retired after an identity-map counterexample. Distributivity (order theory) is related by order vocabulary but governs laws on carriers rather than the all-cover behavior of a map.

Relationships to Other Abstractions¶

Current abstraction Weakly distributive homomorphism Domain-specific

Parents (1) — outgoing typed relations

Weakly distributive homomorphism is a kind of Homomorphism Domain-specific

A weakly distributive join-homomorphism preserves joins and adds an all-cover lifting condition.

Hierarchy path (1) — routes to 1 parentless root

Weakly distributive homomorphism → is a kind of Homomorphism → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Weakly distributive homomorphism sits in a sparse region of the domain-specific corpus (79^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

Ordinary join-homomorphism: preserves joins but may fail the all-cover lift, as the Boolean-target counterexample shows.
Distributive congruence: a property of a congruence on one semilattice in Schmidt's formulation, not this map between two semilattices.
Induced homomorphism: a map arising from a functorial construction; its origin does not establish the lifting law.
Distributivity of a lattice: an equation between meet and join operations on a carrier, not a condition on lifting target covers through a map.

References¶

[1] Friedrich Wehrung, 'A Counterexample to the Congruence Lattice Problem,' author presentation, 7 August 2007, pp. 49–51. https://wehrungf.users.lmno.cnrs.fr/fichiers/07CLP_Pres.pdf registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[2] E. Tamás Schmidt, 'Zur Charakterisierung der Kongruenzverbände der Verbände,' Matematický časopis 18(1), 3–20 (1968), §4, Definitions 4.1 and 4.7, Problem 2. https://dml.cz/bitstream/handle/10338.dmlcz/126757/MathSlov_18-1968-1_1.pdf registry ↩

[3] E. T. Schmidt, 'The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice,' Acta Scientiarum Mathematicarum 43, 153–168 (1981), Definition 3. https://acta.bibl.u-szeged.hu/38622/1/math_043_fasc_001_002.pdf registry ↩