Expansions of the group $Z_8$ (Part I)

TL;DR

The paper studies the lattice of polynomial clones on ${\mathbb Z}_8$ lying between the group polynomial clones and the ring polynomial clones, focusing on the open prime-power case and restricting to polynomials whose nonlinear monomials have even coefficients. It proves a prime-power reduction via a product decomposition when moduli are coprime, and develops a partial description of the interval $\langle P({\mathbb Z}_8,+), M_1 \rangle$ by introducing the clone $M_1$ of ring polynomials preserving a special relation $Z$ and analyzing fully divisible polynomials through a Möbius-basis framework. The main result characterizes the interval as generated by eight infinite families of polynomials (the $2t_n$, $2u_n$, $2v_n$, $2s_n$, $2p_n$, $2q_n$, $2r_n$, $4r_n$) and shows that every clone in the interval is a join of chains built from these families, with a pending algorithm for clone-comparison to complete the classification. This work advances the understanding of clone lattices over the smallest nontrivial prime-power ring ${\mathbb Z}_8$ and sets the stage for a full combinatorial description.

Abstract

We investigate clones in the interval between the group polynomials and the ring polynomials of ${\mathbb Z}_8$. This is the simplest open case of the problem, as the answer is known for ${\mathbb Z}_{p^2}$ (with $p$ prime) and, in general, ${\mathbb Z}_n$ reduces to the case when $n$ is a prime power. The investigated structure proves to be very complicated, so we provide only a partial description. We restrict our attention to polynomials whose nonlinear monomials have even coefficients.

Figures (1)

• Figure 1: The Lattice of Polynomial Clones of $\mathcal{J}_{p^2}$

Theorems & Definitions (27)

• Lemma 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3