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.
Core Idea¶
A weakly distributive homomorphism preserves joins between join-semilattices and lifts every two-piece cover of an image. For μ:S→T, whenever c∈S and a,b∈T satisfy μ(c)≤a∨b, there must be x,y∈S with c≤x∨y, μ(x)≤a, and μ(y)≤b.[1] The condition is universal over all eligible c,a,b.
Preserving joins alone does not guarantee the lift. The map from the two-element chain {0,1} to the four-element Boolean semilattice {0,a,b,1} that sends 0 to 0 and 1 to 1 preserves joins and zero. At c=1 and a∨b=1, however, a source cover of 1 must include source 1, whose image lies below neither atom. This is a direct finite counterexample derived from the definition.
Scope of Application¶
Wehrung gives two classes of canonical (∨,0) maps on compact congruences that are weakly distributive. If A is a reduct of an algebra B on the same carrier, the map from Con_c(A) to Con_c(B) sends a compact congruence to the B-congruence it generates and is weakly distributive.[1] If K is a convex sublattice of L, the analogous map from Con_c(K) to Con_c(L) has the property.[1] The reduct relation and convexity are theorem hypotheses, not optional illustrations.
Schmidt's 1968 work defines a distinct one-carrier distributive-congruence property through a lifting condition.[2] His 1981 formulation of a monomial-congruence join requires the monomial congruences to be weak-distributive; dropping that qualifier changes the statement.[3] Neither one-carrier property is included in this map identity or treated as its alias.
Clarity¶
The name requires more than a distributive target or a join-preserving function. State the source and target explicitly: c,x,y lie in S, while a,b lie in T.[1] Then separate the join-preservation test from the cover-lifting test. A failed triple is enough to exclude a map; a positive proof must cover every triple satisfying the premise.
Manages Complexity¶
The all-cover law packages many individual lifting checks into one typed condition. Wehrung's two theorems certify their respective canonical-map construction classes under a reduct or convex-sublattice hypothesis.[1] They do not license every map between compact-congruence semilattices. The two finite carrier illustrations in the V2 clarify what the canonical maps act on, while the general source theorems supply the all-cover guarantee.
Abstract Reasoning¶
To prove membership, specify S, T, and μ, verify join preservation, and show that every eligible target cover has source witnesses satisfying all three inequalities. To disprove it, produce one target cover without such witnesses. This makes the distinction from ordinary Homomorphism operational. Its strict DAG parent is Homomorphism; Function (Mapping) supplies the broader single-valued map structure through that parent.
Knowledge Transfer¶
The rule transfers literally among join-semilattice maps, including the two compact-congruence constructions above. It should not be transferred merely because another field uses the words “split” or “distribute.” The portable Prime is Function (Mapping), which supplies single-valued assignment; operation preservation belongs to Homomorphism, and the two-piece lift belongs to this domain-specific child.
Relationships to Other Abstractions¶
Current abstraction Weakly distributive homomorphism Domain-specific
Parents (1) — outgoing typed relations
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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 (79th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2769 abstractions)
Nearest neighbors
- Distributive congruence — 0.85
- Monomial congruence — 0.85
- Complete Heyting algebra — 0.83
- Frink ideal — 0.81
- Phragmen–Brouwer theorem — 0.81
Computed from structural-signature embeddings · 2026-10-25
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
[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 ↩