Distributive congruence¶
A join-semilattice congruence whose equivalent inputs can be lifted to cover an ordered equivalent target above their join.
Core Idea¶
In Schmidt's 1968 sense, a distributive congruence is a join-compatible equivalence relation on a join-semilattice that can lift a join within its classes.[1] Suppose x∨y is below u and is equivalent to u. Then we must be able to replace x and y by equivalent elements x₁ and y₁ whose join reaches u or higher.[1] This must work for every such x,y,u.
This property belongs to a congruence on one semilattice. Schmidt also calls a homomorphism distributive when its induced congruence passes the test.[1] That is a derived usage of the relation condition. His later qualified join formulation is kept distinct here.[2]
Scope of Application¶
The definition is used in semilattice and lattice congruence theory. It needs a join operation and the order that join determines; it does not need meets or a greatest element in each equivalence class. Schmidt's Example 1 gives an ideal-generated congruence in a distributive lattice as a positive construction.[1] His Example 2 gives a prime-filter-local extension of a congruence in a relatively complemented distributive lattice.[1]
The qualifier “in Schmidt's 1968 sense” matters. His 1981 Definition 3 uses joins of weak-distributive monomial congruences.[2] His 1968 Problem 2 asks about a qualified decomposition rather than proving an unqualified one.[1] The unqualified claim that any join of monomial congruences characterizes this type is not established by those passages.
Clarity¶
First name the carrier and its join, then prove the relation is an equivalence compatible with join. For the extra lifting test, pick any x,y,u with x∨y≤u and x∨y≡u. The question is whether representatives equivalent to the two inputs can be chosen so their join covers u. Equivalence of the two outputs alone is already part of the premise and does not pass the test.
A monomial congruence asks a different question: does each class have a greatest member? Schmidt defines it separately.[1] Neither test includes the other in general.
Manages Complexity¶
The universal definition can often be checked by sorting the ways its premise can occur. In a three-element chain with classes {0,a} and {1}, the only strict case is 0≤a; replacing one 0 by a reaches the target. In the infinite finite-subset example below, the same construction rule handles every strict case: add missing elements to an input that is already in the upper class. These are explicit proofs, not an assumption that every quotient of a distributive carrier behaves well.
Abstract Reasoning¶
To prove the type, construct suitable replacements for arbitrary ordered congruent targets. To refute it, keep a valid join congruence but find one triple for which both input classes are too small to reach the target. Schmidt's Lemma 4.2 shows that, when the carrier is a distributive semilattice, a distributive congruence has a distributive quotient.[1] The lifting law is the test used here; the quotient result is a consequence under that hypothesis.
Knowledge Transfer¶
Two different positive settings show the same roles. Schmidt's Example 1 identifies an ideal-generated congruence of a distributive lattice as a positive class.[1] For a derived instance, on the chain 0<a<1 collapse the lower ideal {0,a}. The strict premise 0∨0=0≤a lifts to a∨0=a. This particular chain is not a named example in the paper.
Schmidt's Example 2 identifies a prime-filter-local congruence extension as a positive class.[1] For a derived infinite setting, use finite subsets of the positive integers under union. Put all sets containing 1 into one class and leave every other set in its own class. If x∪y lies strictly below an equivalent u, one input contains 1; add the missing finite elements of u to that input. It remains in the same class, and the enlarged union reaches u. Its large class has no greatest finite set, so it is distributive in the 1968 sense but not monomial.
The opposite separation also occurs. Map subsets of {a,b,c} by union onto the five-element diamond M₃, sending singletons to its three atoms and larger sets to its top. The kernel has class tops because the source is finite, yet {a}∨{b}={a,b} is equivalent to {a,b,c} and the singleton input classes cannot be enlarged to reach that upper set. Thus this congruence is monomial but fails the lift. This is a direct counterexample derived from the definition, not one attributed to Schmidt.
Equivalence Relation is the strict Prime parent: every distributive congruence is an equivalence relation, while join compatibility and the lift narrow it. The two-carrier Weakly Distributive Homomorphism and the class-top Monomial Congruence are related identities, not parents.
Relationships to Other Abstractions¶
Current abstraction Distributive congruence Domain-specific
Parents (1) — outgoing typed relations
-
Distributive congruence is a kind of Equivalence Relation Prime
Every distributive congruence is an equivalence relation; join compatibility and Schmidt's 1968 lift make it narrower.
Hierarchy path (1) — routes to 1 parentless root
- Distributive congruence → is a kind of Equivalence Relation
Neighborhood in Abstraction Space¶
Distributive congruence sits in a moderately populated region (56th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Set-Theoretic & Order Structures (54 abstractions)
Nearest neighbors
- Monomial congruence — 0.92
- Distributivity (order theory) — 0.86
- Weakly distributive homomorphism — 0.85
- Completely distributive lattice — 0.84
- Join and meet — 0.84
Computed from structural-signature embeddings · 2026-10-25
References¶
[1] E. Tamás Schmidt, “Zur Charakterisierung der Kongruenzverbände der Verbände,” Matematický časopis 18(1) (1968), 3–20, §4, Definition 4.1 and Examples 1–2 (printed pp. 6–7), Definition 4.7 (p. 9), Problem 2 (p. 10). Full paper: https://dml.cz/bitstream/handle/10338.dmlcz/126757/MathSlov_18-1968-1_1.pdf registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j
[2] E. T. Schmidt, “The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice,” Acta Scientiarum Mathematicarum 43 (1981), 153–168, Definitions 2–3 (printed pp. 156–157). Full issue: https://acta.bibl.u-szeged.hu/38622/1/math_043_fasc_001_002.pdf registry ↩a ↩b