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Open research questions in Advanced Algebra and Logic

150 gap statements mined from Advanced Algebra and Logic papers in our 4.5M-paper local library, which holds 938 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • The paper addresses this gap by counting lattices with five reducible elements.

    Counting of lattices containing up to 5 reducible elements and having nullity up to 3 · 2026 · DOI

  • The realization of ℵ0-monoids is an open problem. There is a lack of understanding of the relationship between ℵ0-monoids and hereditary Von Neumann regular rings.

    Realization of monoids with countable sum · 2026 · DOI

  • Open Problem 1 Are there other families of skew PDSs which can be constructed using unions of cyclotomic classes in Fq? Open Problem 2 Do there exist skew PDSs of types other than Latin square and Paley type - for example, of negative Latin square type? Open Problem 3 We have seen constructions of skew PDSs using bent partitions and cyclotomy.

    Cyclotomic skew partial difference sets and partial difference families · 2026

  • The lack of understanding of the algebraic properties of the asymptotic dynamics of finite-dimensional open quantum systems. The need for a generalization of the results to Schwarz maps.

    Decoherence-free algebras in quantum dynamics · 2026 · DOI

  • It notes that the potential contributions of quasi-set theory to logic are unclear.

    ¿Qué pueden aportar los casi-conjuntos a la lógica? · 2026 · DOI

  • One of the obvious future directions would be to identify the igSPT phases for Rep(H8). This would allow us to discover more fusion categorical anomaly resolutions. Finally, as emphasized in the introduction the story of anomaly resolution and trivially acting symmetries fits nicely in the framework of renormalization group flows.

    SymTFT actions, condensable algebras and categorical anomaly resolutions · 2026 · DOI

  • The lack of a theoretical foundation for IndetermSoft Sets. The need for defined operations for IndetermSoft Sets.

    On Operations of IndetermSoft Sets · 2026 · DOI

  • The eight open questions that the original manuscript left standing have been closed through catalog data, crystal tier census, and algebraic closure operations (§VIII).

    As Above: A Pre-Grammatical Convergent Derivation of the Universal Imscriptive Grammar · 2026 · DOI

  • Pilitowska 7 Questions and remarks In this section, we focus on some open problems and suggest some other APA classes that could be investigated. We do this since the easy computation will reveal a class of APAs that could be studied.

    Associative pentagon algebras · 2026 · DOI

  • The construction is limited to the context of set theory and algebra. The paper does not provide a physical interpretation of the results. The construction is based on the ZFC axioms, which may not be universally accepted.

    A Dual-Directional Cayley-Dickson Construction from the Beth Hierarchy · 2026 · DOI

  • The paper identifies a need for a reference for definitions, theorems, and equations for the AWB.

    GLOSSARIUM AEQUATIONUM. · 2026 · DOI

  • The decidability of the full theory of a given finitely generated free lattice F n (n ≥ 3) was an open problem. The study of the model theory of free lattices was incomplete.

    Elementary properties of free lattices III: Undecidability of the full theory · 2026 · DOI

  • The paper identifies a need to extend known product results for S4 and D4 to weaker systems.

    On Modal Logics of Full Products of Neighborhood Frames · 2026 · DOI

  • The conjecture is a long-standing open problem in combinatorics. The proof requires a combination of structural lemmas and exhaustive computation. The computation is limited to frequencies t ≤ 15.

    Regular union-closed families: a code reduction and the first fifteen rows · 2026 · DOI

  • The proof is restricted to the regular class. The computation is limited to frequencies t ≤ 15.

    Regular union-closed families: a code reduction and the first fifteen rows · 2026 · DOI

  • It remained open whether infinite antichains can always be embedded when the many-one degree is nonrecursive and nonirreducible; Odifreddi stated in a survey 1981 and in his book Classical Recursion Theory in the year 1989 this question explicitly as an open problem.

    Chains and Antichains inside Many-One Degrees and Variants · 2026

  • Future research may focus on further studying the primitive positive constructability poset and its relation to finite simple groups. The authors' techniques may be applied to other problems in universal algebra and constraint satisfaction. Further research may aim to characterize the lower covers of other structures in the primitive positive constructability poset.

    Finite Simple Groups in the Primitive Positive Constructability Poset · 2026 · DOI

  • The authors note that the topmost layers of the poset are known to consist of a single element each, but the lower covers of ℙ₁ are not well understood.

    Finite Simple Groups in the Primitive Positive Constructability Poset · 2026 · DOI

  • The paper does not provide a complete description of complements of maximal sublattices in all semidistributive lattices. The results are limited to certain cases, such as SD lattices and convex geometries of convex dimension 2.

    Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond · 2026 · DOI

  • Which of the properties of ∂F acting on ordinals of finite degree remain true if we consider ∂F acting on countable ordinals of degree at least ω? (2) If the finite condensation is replaced by another condensation, and if we define a multiplication of linear orders based on that condensation, what algebraic structures arise?

    Algebraic Structures Arising from the Finite Condensation on Linear Orders · 2026 · DOI

  • It notes that pseudomonads are already quite complicated, and lax-idempotent pseudomonads are pseudomonads satisfying further properties.

    A Recognition Criterion for Lax-Idempotent Pseudomonads · 2026 · DOI

  • To explore the implications of the construction for the foundations of mathematics. To investigate potential applications of the construction in algebra and geometry. To extend the construction to include non-standard models.

    A Dual-Directional Cayley-Dickson Construction from the Beth Hierarchy · 2026 · DOI

  • Classical Zermelo-Fraenkel Set Theory restricts propositions to strict bivalence. This classical certainty renders ZFC unsuitable for directly modelling phenomena characterized by genuine ambiguity and incompleteness.

    Axiomatization of a Fundamental System for Generalized Ambiguous Four–Degree Membership Function Set Theory · 2026 · DOI

  • The study of the algebraic properties of the asymptotic dynamics of infinite-dimensional open quantum systems. The application of the results to quantum information processing and quantum reservoir engineering.

    Decoherence-free algebras in quantum dynamics · 2026 · DOI

  • Exploring the application of IndetermSoft Sets in various fields. Developing further operations and properties for IndetermSoft Sets.

    On Operations of IndetermSoft Sets · 2026 · DOI

Most-cited papers in Advanced Algebra and Logic

Most recent work

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Open questions raised by papers listed on this page include: (1) There is a lack of understanding of the relationship between ℵ0-monoids and hereditary Von Neumann regular rings (“Realization of monoids with countable sum”, 2026); (2) The lack of understanding of the algebraic properties of the asymptotic dynamics of finite-dimensional open quantum systems (“Decoherence-free algebras in quantum dynamics”, 2026); (3) It notes that the potential contributions of quasi-set theory to logic are unclear (“¿Qué pueden aportar los casi-conjuntos a la lógica?”, 2026); (4) The eight open questions that the original manuscript left standing have been closed through catalog data, crystal tier census, and algebraic closure operations (§VIII) (“As Above: A Pre-Grammatical Convergent Derivation of the Universal Imscriptive…”, 2026); (5) Future research may focus on further studying the primitive positive constructability poset and its relation to finite simple groups (“Finite Simple Groups in the Primitive Positive Constructability Poset”, 2026). 150 gap statements have been mined from Advanced Algebra and Logic papers in our 4.5M-paper local library, which holds 938 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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