Pith. sign in

REVIEW 4 major objections 6 minor 19 references

Associative triple trisystems and standard embeddings

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Associative triple trisystems embed canonically in associative dialgebras, with the three products given by bracketing the two dialgebra products.

desk verdict New ATT definitions and standard embeddings are worth a referee, but the converse of Theorem 13 is underproved as written and Example 5 has a sign error. read the letter →

arxiv 2506.04191 v1 pith:G4LC2C6U submitted 2025-06-04 math.RA

Raúl Felipe , Guillermo Vera de Salas This is my paper · ORCID
classification math.RA MSC 17A3017A6017C99
keywords associativetripletrisystemsdialgebrasJordandisystemsLeibnizsystemsstandardembeddingsdi-endomorphismsKPalgorithm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces associative triple trisystems (ATTs): modules with three trilinear products that generalize associative triple systems by allowing three ways to bracket two dialgebra products $\dashv$ and $\vdash$. It proves that every ATT of the first kind sits inside an associative dialgebra as its standard embedding, with the three products recovered as $x \dashv (y \dashv z)$, $x \vdash (y \dashv z)$, and $x \vdash (y \vdash z)$; the second kind has the analogous embedding into a dialgebra with involution. From any ATT, the same combinations that produce Jordan and Lie triple systems now yield a Jordan triple disystem and a Leibniz triple system, so the classical associative-to-Jordan-and-Lie chain survives at the trisystem level. If correct, the paper supplies the missing associative tier in the trisystem framework and makes dialgebras the natural ambient algebra for studying these structures.

What carries the argument

The KP algorithm is the generating device: it converts a multilinear identity of degree $d$ for one $n$-ary operation into $d$ identities for $n$ new operations indexed by the position of a distinguished central argument, together with interchange identities. Applied to the two classical associativity laws for associative triple systems, it produces the eleven identities defining ATT1 and ATT2. The second carrying object is the module $M(A,A)$ of di-endomorphisms, pairs $(L_\mu(x,y), R_\mu(x,y))$ of left and right multiplication maps, with componentwise products $\dashv$ and $\vdash$. The standard embedding $U(A)=M(A,A)\oplus A$ is assembled from these di-endomorphisms, and its five dialgebra axioms turn out to be exactly the ATT identities re-expressed as bracketings of $\dashv$ and $\vdash$; for ATT2 the same construction is done with four blocks and an involution.

What would settle it

Run a computer algebra search for a finite-dimensional module over a field of characteristic not 2 with three trilinear products satisfying all identities (5)--(15), then check whether the corresponding $M(A,A)\oplus A$ with the block products of Definition 8 satisfies the five dialgebra axioms; one failure would disprove Theorem 13. The analogous check for the four-block construction with identities (19)--(29) would test Theorem 17.

Watch

Extended reading notes

Core claim

The central claim is Theorem 13: a module $A$ with three trilinear products $\{,\!, \}_1$, $\{,\!, \}_2$, $\{,\!, \}_3$ is an associative triple trisystem of the first kind exactly when its standard embedding $U(A)=M(A,A)\oplus A$, built from left and right multiplication di-endomorphisms, is an associative dialgebra; in that dialgebra $\{x,y,z\}_1=x\dashv(y\dashv z)$, $\{x,y,z\}_2=x\vdash(y\dashv z)$, and $\{x,y,z\}_3=x\vdash(y\vdash z)$. Theorem 17 gives the second-kind analogue: an ATT of the second kind is captured by an associative dialgebra with involution on the block module $L(A,A)\oplus A\oplus \bar A\oplus R(A,A)^{\mathrm{op}}$, with products $\{x,y,z\}_1=x\dashv(y^*\dashv z)$, $\{x,y,z\}_2=x\vdash(y^*\dashv z)$, and $\{x,y,z\}_3=x\vdash(y^*\vdash z)$. Along the way, the paper shows that these three products combine to give a Jordan triple disystem and a Leibniz triple system, and it constructs a concrete matrix dialgebra with involution realizing ATT structures outside the classical associative triple system setting.

Load-bearing premise

The load-bearing premise is that the KP algorithm's output, the eleven identities (5)--(15) for the first kind and (19)--(29) for the second, is complete: no further independent identities are needed to define an associative triple trisystem, so the equivalence in Theorem 13 and Theorem 17 captures all such objects.

Editorial extensions

If this is right

  • Every ATT of the first kind is canonically represented inside a dialgebra, so questions about such trisystems can be studied through their enclosing dialgebra $U(A)$.
  • Every associative dialgebra yields an ATT1 by the three bracketed products of Proposition 1, and every dialgebra with involution yields an ATT2 by Proposition 4.
  • Each ATT1 or ATT2 produces a Jordan triple disystem and a Leibniz triple system by explicit formulas, extending the classical associative-to-Jordan and associative-to-Lie correspondences to the di-world.
  • The block-matrix dialgebra construction gives a family of finite-dimensional Leibniz algebras, recovering known four-dimensional examples as low-dimensional members.
  • For the second kind, the standard embedding carries an involution, so ATT2 is governed by dialgebras with involution just as associative triple systems of the second kind are governed by associative algebras with involution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step would be to make the standard embedding functorial: if the construction is natural, ATT homomorphisms should correspond to dialgebra homomorphisms that preserve the distinguished copy of $A$, yielding a category-level equivalence rather than only an object-level one.
  • The block-matrix construction is parametrized by the cut $m_1$, so it likely generates many more isomorphism types of Leibniz algebras than the two examples computed; classifying those algebras for general $m_1$ would test the reach of the construction.
  • Because the ATT identities come from an algorithmic decoration of classical associativity, computer algebra could check whether the eleven axioms are independent or whether a shorter defining set exists; a reduced axiom set would simplify the theory and the embedding proofs.
  • The same di-endomorphism-plus-extra-copy pattern may extend to other ternary di-structures, such as alternative or Malcev triple systems, by applying the same decoration to their defining identities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper introduces two new classes of ternary structures, associative triple trisystems of the first and second kind, obtained by applying the Kolesnikov–Pozhidaev algorithm to the identities of classical associative triple systems. It proves that associative dialgebras (with involution for the second kind) produce examples, that ATT1 and ATT2 give rise to Jordan triple disystems and Leibniz triple systems under the expected triple products, and it constructs concrete examples on matrix algebras. The main structural contribution is a standard embedding: for ATT1, the module U(A)=M(A,A)⊕A is equipped with two products ⊣ and ⊢ and is claimed to be an associative dialgebra if and only if A is an ATT1; for ATT2, a four-component embedding L(A)⊕A⊕Ā⊕R(A) is claimed with an involution. The final sections define di-endomorphisms and use them to build these embeddings.

Significance. If the main theorems are correct, the paper supplies a missing piece of the dialgebra/triple-system analogy and gives explicit, parameter-free constructions that recover the triple products from the embedding. The systematic use of the KP algorithm, the concrete matrix examples, and the explicit definition of di-endomorphisms are useful and go beyond a purely abstract existence statement. The work is, however, computational in character, and the proof of several load-bearing statements is either sketched or internally inconsistent; the significance will be fully realized only after those proofs are completed or verified.

major comments (4)
  1. [§5.3, Theorem 13 (reverse direction)] The 'if' direction of the iff is not proved as written. In the final paragraph the axioms (5)–(15) are discharged by citing Proposition 12 and specific forward cases such as 3(a) and 7(a), but those facts were established under the assumption that A is an ATT1. In the converse one may only assume the five dialgebra identities on U(A). For example, deriving (5) requires both {x,{y,z,u}_1,v}_1 = {x,y,{z,u,v}_1}_1 and {{x,y,z}_1,u,v}_1 = {x,y,{z,u,v}_1}_1; the first of these needs the identification (x◁y)⊣(z◁u)=x◁{y,z,u}_1 from Proposition 12(1), which is not available from the dialgebra axioms alone. The same gap affects (7), (8), (12), and (13), whose listed derivations also invoke Proposition 12. The equivalence is therefore incomplete.
  2. [§4, Example 5] The claimed bracket [E3,X]=E3 is a sign error. With [A,B]=A⊣B−B⊢A and the block definitions in Example 4, E3⊣X=0 and X⊢E3=E3, hence [E3,X]=−E3. The subsequent identification L≅L23(0,1) is therefore not supported by the displayed relations; the example should be recomputed and the reference to the classification in [5] adjusted.
  3. [§3, Theorem 3; §4, Theorems 5–6] The verification of the Leibniz triple system axioms is not self-contained. In the proof of Theorem 3 the text states that 'for two arbitrary terms, at least two elements occupy different places, it follows ... that none of terms can be canceled,' and then immediately presents a table of 48 cancellations t_{i,j}=−t_{k,l}. This is internally contradictory, and the listed cancellations are asserted without showing which of the identities (5)–(15) justifies each one. Since Theorems 5 and 6 reuse the same table with modified rows and no further derivation, the central claims that ATT1 and ATT2 yield Leibniz triple systems rest on unverified computation rather than on a checkable proof.
  4. [§5.4, Proposition 15] Items 1, 4, 5, and 8 of Proposition 15 are stated 'for every ... i∈{1,2,3}', yet the left-hand sides do not depend on i whereas the right-hand sides contain L◁(x,{u,z,y}_i) or L▷(x,{u,z,y}_i). As stated this would force {u,z,y}_1={u,z,y}_2={u,z,y}_3, which is not among the axioms of Definition 5. The index i must be fixed to a definite value (or its dependence must be proved), and the corresponding equalities must be derived from (19)–(29). Without this, Theorem 17, which relies on Proposition 15, is not established.
minor comments (6)
  1. [Abstract] The word 'standar' in 'standar embedding' should be 'standard'.
  2. [Section 4 heading] The heading should read 'Associative triple trisystems of the second kind' rather than 'Associative triple systems of second kind'.
  3. [Theorem 13, case 5(a)] The right-hand side contains {z,y,v}_2, which appears to be a typo for {z,u,v}_2.
  4. [Theorem 6, first sentence] 'preovious' should be 'previous'; the sentence also has a typo 'preovious theorem'.
  5. [Definition 4 and following paragraph] The decomposition A=Aann⊕A uses A for both the whole space and a subspace; a different letter for the subspace would avoid confusion.
  6. [Example 4, final remark] The statement that the constructed ATT2 'is not arised from an associative dialgebra' is unclear, since Theorem 17 later embeds every ATT2 into a dialgebra with involution; the intended meaning should be clarified.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 13's converse derives the ATT axioms from Proposition 12 and forward-direction cases whose proofs already assume those axioms.

  1. other [Theorem 13, Section 5.3, final paragraph of the proof ('Reciprocally...')]
    "Reciprocally, if U(A) is an associative dialgebra, we recover the relations • (5) from 3(a) and 7(a), • (6) from 4(a) and 8(a), • (7) from Proposition 12(2), • (8) from 5(d) and 12(b), ... • (12) from Proposition 12 (2-3), • (13) from Proposition 12(1-2), • (14) from 3(c) and 5(c), • (15) from 4(c) and 6(c)."

    In the forward part of this same proof, items 3(a), 7(a), 4(a), 8(a), 3(c), 5(c), etc. are established only after invoking Proposition 12, whose statement begins 'Let A be an associative triple trisystem of the first kind' and whose proof uses the ATT axioms (5)-(15). In the converse, the hypothesis is only that U(A) is an associative dialgebra; Proposition 12 and those forward case proofs are not available unless the axioms being recovered have already been proved. E.g., 3(a) by itself, evaluated without Proposition 12, gives {x,y,{z,u,v}3}1 = {x,y,{z,u,v}1}1 (a fragment of (14)), not (5). Thus the reverse implication derives (5)-(15) from statements whose proofs already assume (5)-(15); the iff is not independently established as written.

full rationale

Apart from the converse of Theorem 13, the paper's construction is self-contained: Definitions 4 and 5 are produced by the external Kolesnikov-Pozhidaev algorithm from the identities (1) and (2), Proposition 1 and Proposition 4 verify that associative dialgebras (with involution) give ATTs by direct computation, and Theorems 2, 3, 5 and 6 check the JTD and LeibTS axioms from the ATT axioms without importing the target result. The standard embedding is built explicitly, and the forward direction of Theorem 13 (if A is ATT1 then U(A) is a dialgebra) is a long but direct verification. The circularity is localized to the reverse direction: the bullet list at the end of Theorem 13 'recovers' each ATT identity from Proposition 12 and from enumerated forward cases, but those cases were proved under the assumption that A is an ATT1. Since the converse is exactly the claim that the five dialgebra identities on U(A) force those ATT identities, invoking Proposition 12 and the forward cases is invoking the conclusion. This is a genuine circular proof step in the central equivalence theorem, though it does not affect the definitions, the forward embeddings, or the JTD/LeibTS consequences.

Assumptions & free parameters 0 free parameters · 5 assumptions · 4 invented entities

The paper introduces no fitted numerical parameters. Its axioms are standard background facts: the base ring contains 1/2, the KP algorithm is accepted as a valid transformation, and prior definitions of JTD, LeibTS, dialgebras, and involutions are taken as given. The new structures (ATT1, ATT2, di-endomorphisms, standard embedding) are defined explicitly and supported by direct proofs rather than by circular assumptions.

assumptions (5)
  • domain assumption The scalar ring φ contains 1/2.
    Stated at the start of Section 2; needed for Jordan-type products and the classical Jordan algebra construction.
  • standard math The KP algorithm as stated in Definition 1 is a valid transformation of identities.
    Background tool from Kolesnikov and Pozhidaev, recalled in [2]; the paper relies on it to derive ATT identities.
  • standard math The definitions of Jordan triple disystem and Leibniz triple system from [2] and [3] are used as given.
    Recalled in Definitions 2 and 3; the paper proves that ATTs satisfy these, without altering their axioms.
  • standard math Associative dialgebras and the notion of involution from [18] are taken as known.
    The paper builds directly on Loday's dialgebras and the involution concept from Felipe's earlier work.
  • standard math The classical standard embedding for associative triple systems (Meyberg [16]) serves as motivation.
    The new di-endomorphism construction is modeled on this known embedding, but the proof is self-contained.
invented entities (4)
  • Associative triple trisystem of the first kind (ATT1) independent evidence
    purpose: New variety of triple trisystems defined by applying the KP algorithm to ATS1 identities; the central object of the paper.
    Theorem 13 shows equivalence with the standard embedding being an associative dialgebra; this gives an independent structural characterization beyond the definition.
  • Associative triple trisystem of the second kind (ATT2) independent evidence
    purpose: Analogous variety from ATS2 identities together with an involution; yields JTD and LeibTS.
    Proposition 4 constructs ATT2 from any dialgebra with involution, and the matrix example provides explicit realizations.
  • Di-endomorphisms of a module (DiEnd(D)) independent evidence
    purpose: Pairs of endomorphisms with two compositions ⊣ and ⊢ forming an associative dialgebra; used to build standard embeddings.
    Proposition 7 proves directly that these products form an associative dialgebra, a checkable construction independent of the embedding results.
  • Standard embedding U(A) = M(A,A) ⊕ A independent evidence
    purpose: Canonical dialgebra into which every ATT embeds.
    Theorems 13 and 17 prove the equivalence explicitly; the construction is detailed and checkable from the definitions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Associative triple trisystems and standard embeddings." pith.science (2026). https://pith.science/paper/G4LC2C6U

@misc{pith2026250604191,
  author       = {Pith},
  title        = {Pith review of: Associative triple trisystems and standard embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4LC2C6U}},
  note         = {Machine review of arXiv:2506.04191}
}
read the original abstract

Building on the established theories of Jordan triple disystems and Leibniz triple systems, we introduce and develop the theory of associative triple trisystems, filling a significant gap in the existing framework. We establish the classical relationships between associative, Jordan, and Lie triple systems within the context of trisystems. We present a significant example by equipping the space of matrices with a non-trivial associative dialgebra structure. We conclude defining the concept of di-endomorphisms of any module, which enables the construction of the standard embedding for any associative triple trisystem.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [5]

    E. M. Ca˜ nete and A. Kh. Khudoyberdiyev The classification of 4- dimensional Leibniz algebras. Linear Algebra and its Applications. 439(1) (2013), 273-288

    work page 2013

  2. [1]

    Ayupov, B.A

    Sh.A. Ayupov, B.A. Omirov, On Leibniz algebras, in: Algebra and Operator Theory (Tashkent, 1997), Kluwer Acad. Publ., Dordrecht, (1998), 1–12

    work page 1998

  3. [2]

    M. R. Bremner, R. Felipe and J. S´ anchez-Ortega Jordan triple disystems. Computers and Mathematics with Applications 63 (2012), 1039-1055

    work page 2012

  4. [3]

    M. R. Bremner and J. S´ anchez-Ortega Leibniz triple systems. Communica- tions in Contemporary Mathematics 16 (2014)

    work page 2014

  5. [4]

    M. R. Bremner. On the definition of quasi-Jordan algebra. Communications in Algebra, 38(12) (2010), 4695-4704

    work page 2010

  6. [6]

    Casas and M.A

    J.M. Casas and M.A. Insua and M. Ladra and S. Ladra An algorithm for the classification of 3-dimensional complex Leibniz algebras. Linear Algebra and its Applications 436(9) (2012), 3747-3756

    work page 2012

  7. [7]

    Cuvier, Alg` ebres de Leibnitz: d´ efinitions, propri´ et´ es.Ann

    C. Cuvier, Alg` ebres de Leibnitz: d´ efinitions, propri´ et´ es.Ann. Sci. ´Ecole Norm. Sup. 27 (1994), 1-45

    work page 1994

  8. [8]

    Felipe, A brief foundation of the left-symmetric dialgebras

    R. Felipe, A brief foundation of the left-symmetric dialgebras. Comunicaci´ on del CIMAT No I-11-02/18-03-2011 (2011)

    work page 2011

Show all 19 references
  1. [9]

    A. V. Gnedbaye Jordan triples and operads. Journal of Algebra 231 (2000), 744-757

    2000

  2. [10]

    Gubarev and P

    V.Y. Gubarev and P. S. Kolesnikov. The Tits–Kantor–Koecher Construc- tion for Jordan Dialgebras . Communications in Algebra, 39(2) (2011), 497–520

    2011

  3. [11]

    P. S. Kolesnikov, Varieties of dialgebras and conformal algebras. Siberian Mathematical Journal, 49 (2008), 257-272

    2008

  4. [12]

    J. L. Loday, Une version non commutative des alg` ebres de Lie: les alg` ebres de Leibniz, Ens. Math. 39 (1993), 269-293

    1993

  5. [13]

    J. L. Loday, Dialgebras., in: Dialgebras and related operads, Lectures Notes in Mathematics, vol. 1763, Springer Verlag, 2001, 7-66

    2001

  6. [14]

    J. L. Loday and T. Pirashvili. Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann., 296(1) (1993), 139-158

    1993

  7. [15]

    Loos, Assoziative tripelsysteme , Manuscripta Mathematica 7 (1972), 103–112

    O. Loos, Assoziative tripelsysteme , Manuscripta Mathematica 7 (1972), 103–112. 47

    1972

  8. [16]

    Meyberg, Lectures on Algebras and Triple Systems, Lecture notes, Char- lottesville (V A): University of Virgina (1972)

    K. Meyberg, Lectures on Algebras and Triple Systems, Lecture notes, Char- lottesville (V A): University of Virgina (1972)

    1972

  9. [17]

    A. P. Pozhidaev, 0 -dialgebras with bar-unity, Rota-Baxter and 3-Leibniz algebras. Contemporary Mathematics 499 (2009), 245-256

    2009

  10. [18]

    Felipe, An analogue to functional analysis in dialgebras

    R. Felipe, An analogue to functional analysis in dialgebras. International Mathematical Forum. 2 (2007), 1069-1091

    2007

  11. [19]

    Vel´ asquez and R

    R. Vel´ asquez and R. Felipe. Quasi-Jordan algebras. Communications in Algebra, 36(4) (2008), 1580-1602. 48

    2008

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.