Idempotent-Separating Representations

• Idempotent-separating representations are techniques that decompose rings and valuation systems into orthogonal idempotents, enabling efficient structural analysis.
• They facilitate the reduction of parameterized linear difference equations by transforming global problems into parallel subproblems over local domains.
• Algorithmic strategies using infimum and supremum representations optimize operations in algebraic manipulations and inference, balancing computational cost.

Idempotent-separating representations arise in both the algebraic and logical analysis of systems featuring idempotent decompositions. These representations enable the decomposition and efficient manipulation of mathematical objects—such as elements of rings with zero-divisors or valuations in information systems—by exploiting the structural properties conferred by idempotency. In particular, they underpin reduction strategies for solving parameterized linear difference equations (PLDEs) in rings with idempotent splittings, as well as canonical expansions and inference algorithms in valuation-based systems (VBS) with idempotent combination rules (Ablinger et al., 2021, Hernandez et al., 2013).

1. Idempotent Decompositions and Their Structural Properties

In a commutative ring $R$ with unity, an element $e\in R$ is idempotent if $e^2 = e$. A finite set of pairwise orthogonal idempotents $\{e_1,\dots,e_r\}$ is complete when $e_i^2=e_i$, $e_ie_j=0$ for $i\neq j$, and $\sum_{i=1}^r e_i=1$. Any $f\in R$ admits a unique decomposition $f = e_1f +\dots+e_rf$. This makes $R$ isomorphic to the direct sum $e_1R\oplus\dots\oplus e_rR$. Each $e_iR$ is a subring with unit $e_i$. If each $e_iR$ is an integral domain, $R$ is a direct sum of domains, but generally has zero-divisors across components (Ablinger et al., 2021).

In valuation-based systems, idempotency refers to the combination operation: for a set of valuations $V$, the system is idempotent if $V\otimes V=V$ for all $V$. Such systems possess a lattice structure where the binary combination is the join operation. Idempotent systems include Boolean set algebras and convex-set-valued valuations (Hernandez et al., 2013).

2. Idempotent-Separating Representations: Definitions and Canonical Forms

Given the lattice structure induced by idempotency, each element can be decomposed via either an infimum (meet) or a supremum (join) of basic generators.

• Infimum (Meet) Representation: Every valuation $V$ can be written as $V = \wedge_{W\in M_V} W$, where $M_V$ is a (typically minimal) set of "most informative" idempotent valuations above $V$. A lower representation system (LRS) $L^*$, closed under liftings, generates all such $M_V$ uniquely: removal of any $W$ destroys the equality.
• Supremum (Join) Representation: Dually, $V = \vee_{u\in N_V} u$, with $N_V$ a set of "least informative" idempotent elements below $V$. An upper representation system $U^*$, minimal for coverage, guarantees this expansion (Hernandez et al., 2013).

Within difference rings split by idempotents, every element $y$ can be written as $y = \sum_{i=1}^r e_i y_i$ with $y_i\in e_iR$. Each $e_iR$ acts as an indecomposable "generator" for the representation, and manipulating the original element reduces to parallel manipulations on each $y_i$.

3. The Idempotent-Separating Reduction in Difference Rings

For $R$ a difference ring split as $R = e_1R\oplus\dots\oplus e_rR$, parameterized linear difference equations of the form

$$a_0 y + a_1 \sigma(y) + \dots + a_m \sigma^m(y) = c_1 f_1 + \dots + c_d f_d$$

with coefficients in $R$ allow a systematic reduction. Setting $y = \sum_{i=1}^r e_i y_i$ and considering the cyclic action of the automorphism $\sigma$, one constructs a shifted-projected system whose solution space separates as $r$ parallel problems, each over the local integral domain $e_iR$. More precisely, for each component, the PLDE reduces to

$L_i(y_i) = \varphi_i$

with $L_i$ a difference operator using the $r$-fold shift on $e_iR$, and $\varphi_i\in e_iR$. Existing solvers for classical PLDEs can be applied to each, after which global solutions are reassembled by coordinating the intersection of solution spaces on the constant field (Ablinger et al., 2021).

4. Algorithmic Strategies and Complexity Considerations

Algorithmic procedures bifurcate based on the choice of infimum (meet) or supremum (join) representations.

• Infimum Representation: Combination (join) of valuations corresponds to set union of their $M$-sets. Marginalization (projection) requires identifying minimal subsets whose meet eliminates a variable ("deletion-dimension"), which can be computationally intensive.
• Supremum Representation: Marginalization is implemented as direct projection of elements in $N$-sets, while combination (meet/intersection) necessitates enumeration of minimal consistent pairs, which can be exponential in the number of generators. These representational dualities inform the choice of algorithm depending on whether the dominant computational expense lies in repeated joins or projections (Hernandez et al., 2013).

For difference rings, after splitting the global system and solving the $r$ component PLDEs, a compatibility check is imposed at the level of constants. This is performed via finite linear algebra over the constant subfield. Reassembly ensures that all solutions of the original PLDE are captured without redundancy (Ablinger et al., 2021).

5. Concrete Examples: PLDEs and Valuation Systems

In the context of difference rings with idempotent decomposition, a canonical example is provided by $R = \mathbb{Q}(x)[y][s][\bar{s}]$ with $y^2=1$, leading to two idempotents $e_0 = (1-y)/2$, $e_1=(1+y)/2$. Solving a three-term recurrence in $R$ involves:

1. Projecting onto $e_0R$ and $e_1R$ to obtain two order-two PLDEs,
2. Solving each using integral-domain-oriented algorithms (e.g., telescoping/nested sum solvers),
3. Recombining the solutions with compatible constants determined by intersection (Ablinger et al., 2021).

For valuation-based systems, classical cases include:

• Finite sets: Lower system $L^*$ comprises complements of singletons, upper system $U^*$ comprises singletons; representation mirrors propositional logic and set partitioning.
• Convex polytopes: Lower system $L^*$ is all half-spaces; upper system $U^*$ the set of extreme points. Deletions and combinations correspond to geometric operations such as hyperplane resolution and convex-hull computation (Hernandez et al., 2013).

System	Lower Representation $L^*$	Upper Representation $U^*$
Finite Sets	Complements of singletons	Singletons
Convex Polytopes	Half-spaces (inequalities)	Extreme points

6. Applications, Implementations, and Implications

The idempotent-separating approach is foundational for the reduction of PLDEs with coefficients in rings featuring nested sums, products, and roots of unity (i.e., $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields). Solutions and reductions can thus be performed atomically in each domain component before reassembly, leveraging constant-stable difference field solvers. This methodology is implemented in symbolic computation packages such as RISC–Sigma/HarmonicSums (Ablinger et al., 2021).

In valuation-based systems, idempotent-separating representations enable both canonical expansions (unique minimal decomposition into basic idempotents) and efficient inference, particularly in knowledge representation and reasoning on sets or polytopes. Algorithmic choices—between infimum and supremum strategies—enable adaptation to the dominant computational cost structure (e.g., repeated joins versus repeated projections) (Hernandez et al., 2013).

A plausible implication is that such representations, by isolating indispensable idempotent generators, provide not only theoretical clarity (uniqueness, canonicity, absence of redundancy) but also algorithmic leverage for both algebraic and logical/symbolic computation.

(Ablinger et al., 2021): https://arxiv.org/abs/([2102.03307](/papers/2102.03307), Hernandez et al., 2013): https://arxiv.org/abs/([1302.1546](/papers/1302.1546))