Four generators of an equivalence lattice with consecutive block counts

Abstract

The block count of an equivalence $μ\in$ Equ$(A)$ is the number blnum$(μ)$ of blocks of (the partition corresponding to) $μ$. We say that $X=\{μ_1,μ_2,μ_3,μ_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(μ_{i+1})$ = blnum$(μ_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.

Figures (4)

• Figure 1: Zádori's construction for $|A|=19$
• Figure 2: Right-going and left-going sequences
• Figure 3: Generating a quasiorder lattice
• Figure 4: Illustrating the proof of Lemma \ref{['lemma:tstp']}

Theorems & Definitions (16)

• Lemma 1.1: Kulin's Lemma
• Claim 1.2: zadori; exemplifying the odd case of Zádori's construction
• proof
• Lemma 1.3
• Claim 1.4
• proof : Outline of the proof
• Theorem 2.1
• Remark 2.2
• Definition 2.3: czg4ghoriz
• Lemma 2.4