[Paper Review] On the lattice of equational classes of Boolean functions and its closed intervals

Miguel Couceiro|arXiv (Cornell University)|May 17, 2011

Advanced Algebra and Logic11 references17 citations

TL;DR

This paper investigates the uncountable closed intervals in the lattice of equational classes of Boolean functions, establishing a necessary and sufficient condition for an interval $[\mathcal{C}_1, \mathcal{C}_2]$ to be uncountable: the difference set $\mathcal{C}_2 \setminus \mathcal{C}_1$ must contain an infinite antichain under the pre-order $\preceq_{\bf V}$. The authors classify all such intervals in the Boolean case and show that uncountability arises precisely when non-associative functions are present in the difference set.

ABSTRACT

Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boolean case |A|=2, the lattice E_A contains uncountably many equational classes, but only countably many of them are clones. The aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its "closed" intervals" [C_1,C_2], for clones C_1 and C_2. For |A|=2, we give a complete classification of all such closed intervals in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of E_A.

Motivation & Objective

• To understand the structure of the uncountable lattice of equational classes of Boolean functions, which is finer than the lattice of clones.
• To classify all closed intervals $[\mathcal{C}_1, \mathcal{C}_2]$ in the lattice $\mathbf{E}_{\mathbb{B}}$ of equational classes of Boolean functions based on their cardinality.
• To determine a necessary and sufficient condition for such intervals to be uncountable, focusing on idempotent classes $\mathcal{C}_1$ and $\mathcal{C}_2$.
• To analyze the role of non-associative Boolean functions and infinite antichains in distinguishing uncountable from countable intervals.

Proposed method

• Define the lattice $\mathbf{E}_A$ of equational classes of operations on a finite set $A$, with $|A| \geq 2$, using class composition and functional equations.
• Introduce the pre-order $\preceq_{\bf V}$ on operations, which captures variable substitution equivalence, and use it to define antichains in the difference set $\mathcal{C}_2 \setminus \mathcal{C}_1$.
• Construct explicit infinite antichains of Boolean functions using families such as $f_n$, $g_n$, $u_n$, $t_n^u$, $H_n$, $G_m^n$, $T_n$, and $s_n$, each demonstrating non-associativity.
• Prove that an interval $[\mathcal{C}_1, \mathcal{C}_2]$ is uncountable if and only if $\mathcal{C}_2 \setminus \mathcal{C}_1$ contains an infinite antichain under $\preceq_{\bf V}$.
• Use the characterization to classify all closed intervals in $\mathbf{E}_{\mathbb{B}}$ by size, distinguishing countable and uncountable cases.
• Show that every uncountable interval contains a minimal uncountable interval, and that such intervals must contain a non-associative function in $\mathcal{C}_2 \setminus \mathcal{C}_1$.

Experimental results

Research questions

• RQ1 What characterizes the uncountable closed intervals in the lattice $\mathbf{E}_{\mathbb{B}}$ of equational classes of Boolean functions?
• RQ2 How does the presence of non-associative functions in $\mathcal{C}_2 \setminus \mathcal{C}_1$ relate to the uncountability of the interval $[\mathcal{C}_1, \mathcal{C}_2]$?
• RQ3 Can the uncountable intervals in $\mathbf{E}_{\mathbb{B}}$ be completely classified based on structural properties of the difference set $\mathcal{C}_2 \setminus \mathcal{C}_1$?
• RQ4 Is there a necessary and sufficient condition for an interval $[\mathcal{C}_1, \mathcal{C}_2]$ to be uncountable, in terms of the existence of infinite antichains under $\preceq_{\bf V}$?
• RQ5 What is the role of duality and associativity in distinguishing between countable and uncountable intervals in $\mathbf{E}_{\mathbb{B}}$?

Key findings

• The lattice $\mathbf{E}_{\mathbb{B}}$ of equational classes of Boolean functions contains uncountably many elements, though only countably many are clones.
• An interval $[\mathcal{C}_1, \mathcal{C}_2]$ in $\mathbf{E}_{\mathbb{B}}$ is uncountable if and only if $\mathcal{C}_2 \setminus \mathcal{C}_1$ contains an infinite antichain under the pre-order $\preceq_{\bf V}$.
• The authors construct explicit infinite antichains of Boolean functions, such as $f_n$, $g_n$, $u_n$, $t_n^u$, $H_n$, $G_m^n$, $T_n$, and $s_n$, each demonstrating non-associativity.
• Every uncountable closed interval in $\mathbf{E}_{\mathbb{B}}$ contains a minimal uncountable interval, and such intervals must contain at least one non-associative function in $\mathcal{C}_2 \setminus \mathcal{C}_1$.
• The presence of a non-associative function in $\mathcal{C}_2 \setminus \mathcal{C}_1$ is both necessary and sufficient for the interval $[\mathcal{C}_1, \mathcal{C}_2]$ to be uncountable.
• The classification of closed intervals in $\mathbf{E}_{\mathbb{B}}$ is fully determined by the existence of infinite antichains in $\mathcal{C}_2 \setminus \mathcal{C}_1$, providing a complete structural characterization.