Finite Simple L-Algebras

• Finite Simple L-Algebras are finite algebraic structures with a distinguished unit and binary operation satisfying specific axioms, ensuring no nontrivial ideals.
• The classification theorem shows that every such algebra is isomorphic to a unique linear CKL-algebra Aₙ, constructed via a strict linear order and defined multiplication rules.
• Their spectral properties and role in semidirect product constructions provide foundational insights that extend to related areas like Hilbert algebras in algebraic logic.

A finite simple L-algebra is a finite algebraic structure equipped with a distinguished unit and a binary operation subject to specific axioms, with no nontrivial ideals. The complete classification and structure of these objects, as well as their relationship to certain subclasses of CKL-algebras and Hilbert algebras, have been established and provide foundational insight into semidirect product constructions and spectral properties in algebraic logic (Properzi et al., 9 Dec 2025).

1. Definition and Axioms of L-Algebras

An L-algebra is a set $X$ with a distinguished element $1$ and a bilinear operation "$\cdot$" satisfying the following for all $x, y, z \in X$:

• (LA1): $1 \cdot x = x$, $x \cdot 1 = x \cdot x = 1$
• (LA2): $$(x \cdot y) \cdot (x \cdot z) = (y \cdot x) \cdot (y \cdot z)$$
• (LA3): $x \cdot y = y \cdot x = 1 \implies x = y$

A two-sided ideal $I \subseteq X$ is a subset such that $x \in I \implies y \cdot x \in I$ for all $y \in X$, and whenever $x \in I$ and at least one of $x \cdot y\in I$, $(x \cdot y)\cdot y\in I$, or $y \cdot(x \cdot y)\in I$ holds, then $y \in I$. $X$ is simple if its only ideals are $\{1\}$ and $X$ [(Properzi et al., 9 Dec 2025), Def. 2.1/2.2].

Extensions, including KL-, CKL-, and Hilbert algebras, impose additional identities (e.g., for CKL: $x \cdot (y \cdot z) = y \cdot (x \cdot z)$; for Hilbert: $$x \cdot (y \cdot z) = (x \cdot y) \cdot (x \cdot z)$$).

2. Classification Theorem for Finite Simple L-Algebras

All finite simple linear L-algebras are classified up to isomorphism by a family $\mathcal{F} = \{A_n : n \geq 1 \}$, where each $A_n$ is the unique linear CKL-algebra of size $n$ with no proper ideals. Formally:

Theorem: Let $X$ be a finite simple linear L-algebra of cardinality $n$. Then $X \cong A_n$. Conversely, each $A_n$ is simple [(Properzi et al., 9 Dec 2025), Thm. 5.4, 5.12].

In the subclass of finite "tail${}^+$" CKL-algebras, this family $\mathcal{F}$ comprises all simple examples, providing a complete classification within this context.

3. Construction of the Family $\mathcal{F} = \{A_n\}$

For each integer $n \geq 1$, $A_n = \{x_0, x_1, \ldots, x_{n-1}\}$ carries a strictly linear order $x_0 > x_1 > \cdots > x_{n-1}$, and the multiplication is given by

$x_i \cdot x_j = x_{\max(j-i, 0)}$

Equivalently, for $i<j$, $x_i \cdot x_j = x_{j-i}$; for $i \geq j$, $x_i \cdot x_j = x_0 = 1$ (the unit). $A_n$ is the unique linear CKL-algebra on $n$ points which has no nontrivial ideals [(Properzi et al., 9 Dec 2025), §3].

Illustrative Small Examples:

	$x_0$	$x_1$	$x_2$
$x_0$	$x_0$	$x_1$	$x_2$
$x_1$	$x_0$	$x_0$	$x_1$
$x_2$	$x_0$	$x_0$	$x_0$

Multiplication table for $A_3 = \{x_0 > x_1 > x_2\}$.

4. Structure of Ideals and Simplicity

Any proper ideal in a linear L-algebra takes the form $I = \{ x \in X \mid x \geq x_i \}$, where $x_{i+1}$ is invariant (i.e., $x$ such that $y \cdot x_{i+1} = x_{i+1}$ for all $y$). The simplicity condition restricts all but $x_1, \ldots, x_{n-1}$ from being invariant, so the only possible ideals are $\{1\}$ and $X$ [(Properzi et al., 9 Dec 2025), Lem. 4.3, Thm. 5.4].

The inductive construction: to extend a simple linear L-algebra of size $n-1$ to size $n$, introduce $x_{n-1}$ with $x_1 \cdot x_{n-1} = x_{n-2}$, uniquely yielding $A_n$. This directly shows the rigidity and uniqueness of the classification.

5. Coincidence with Simple Tail${}^+$ CKL-Algebras

A tail${}^+$ CKL-algebra either consists of a single linear "tail" above a minimal element or is formed by adjoining a minimal element to a smaller tail. The main result states that any simple tail${}^+$ CKL-algebra is forced to be linear and thus must coincide with one of the $A_n$ [(Properzi et al., 9 Dec 2025), Thm. 5.12]. Thus, within this fundamental subclass, $\mathcal{F}$ exhausts all simple structures.

6. Spectral Structure and Semidirect Products

An L-algebra's spectrum $\mathrm{Spec}(X)$ is the set of its prime ideals, endowed with a Zariski-style topology. For semidirect product constructions $X \rtimes_\rho Y$, the spectrum decomposes as

$$\mathrm{Spec}(X \rtimes Y) \cong \rho\text{-}\mathrm{Spec}(X) \sqcup \mathrm{Spec}(Y)$$

This framework provides a precise analysis of ideal and prime ideal structures in complex semidirect products of L-algebras [(Properzi et al., 9 Dec 2025), §4].

7. Applications to Linear Hilbert Algebras

Finite simple Hilbert algebras have size at most 2. More generally, any finite linear Hilbert algebra $X$ of size $n$ is isomorphic to the "Hilbert chain" $LH_n = \{x_0 > x_1 > \cdots > x_{n-1}\}$ with operation: $$x_i \cdot x_j = \begin{cases} 1 & \text{if } i \geq j, \ x_j & \text{if } i < j. \end{cases}$$ Every such $LH_n$ arises as a symmetric semidirect product of a smaller Hilbert algebra with $A_2$, using the action of $A_2$ that is the identity above a cut-point and a constant at 1 on the rest. This recursive structure is controlled by the classification of finite simple L-algebras [(Properzi et al., 9 Dec 2025), Prop. 6.6, §6].