The Lattice of Coalition Logics between MCL and CL: Coatoms and Compact Meets
Abstract
For a fixed finite set N of at least two agents, we investigate the lattice LN of coalition logics between minimal coalition logic (MCL) and classical coalition logic (CL), with extensions required to be structural. This lattice is order-dual to an interval in the subvariety lattice of Boolean coalition algebras. The eight basic extensions obtained by imposing seriality, independence of agents, and determinism form a Boolean cube under inclusion, but not a sublattice of LN. Our main results concern coatoms and compact meets. First, finite critical coalition algebras, realized as complex algebras of finite general concurrent games, yield three infinite families of pairwise distinct coatoms containing seriality, distinguished by failures of independence, determinism, or both. An explicit splitting characterizes the unique coatom omitting seriality. Second, the meet of two basic extensions is compact if and only if the extensions are comparable. For incomparable pairs, no axiomatization of the meet can use formulas of uniformly bounded modal depth, even with infinitely many axioms. Consequently, LN and three natural principal intervals are non-coherent algebraic lattices. These results reveal lattice-theoretic phenomena beyond the inclusion order of the eight basic extensions.
Related
- Classical and Quantum Logics with Multiple Non-Equivalent and Non-Isomorphic Models
- Logics of Filter Bubbles
- Uncountably many maximally consistent neighborhood complete congruential modal logics
- Cooperative Integer Programming Games: Core Stability and Optimal Coalition Structures
- Closure Semantics for Weak Modal Logics: Completeness and Representations
BibTeX
@misc{wang2026lattice,
title = {{The Lattice of Coalition Logics between MCL and CL: Coatoms and Compact Meets}},
author = {Shanxia Wang and Yuqiang Dou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24466},
eprint = {2610.24466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24466},
}
RIS
TY - UNPB
TI - The Lattice of Coalition Logics between MCL and CL: Coatoms and Compact Meets
AU - Shanxia Wang
AU - Yuqiang Dou
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24466
UR - https://arxiv.org/abs/2610.24466
AB - For a fixed finite set N of at least two agents, we investigate the lattice L_N of coalition logics between minimal coalition logic (MCL) and classical coalition logic (CL), with extensions required to be structural. This lattice is order-dual to an interval in the subvariety lattice of Boolean coalition algebras. The eight basic extensions obtained by imposing seriality, independence of agents, and determinism form a Boolean cube under inclusion, but not a sublattice of L_N. Our main results concern coatoms and compact meets. First, finite critical coalition algebras, realized as complex algebras of finite general concurrent games, yield three infinite families of pairwise distinct coatoms containing seriality, distinguished by failures of independence, determinism, or both. An explicit splitting characterizes the unique coatom omitting seriality. Second, the meet of two basic extensions is compact if and only if the extensions are comparable. For incomparable pairs, no axiomatization of the meet can use formulas of uniformly bounded modal depth, even with infinitely many axioms. Consequently, L_N and three natural principal intervals are non-coherent algebraic lattices. These results reveal lattice-theoretic phenomena beyond the inclusion order of the eight basic extensions.
ER -