{
  "abstract": "We introduce the $\\mathfrak M$arseille anti-construction $\\mathfrak{Ma}$ for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety $\\Var$, a new variety $\\mathfrak{Ma}$-$\\Var$ that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric $\\big($pre-Jacobi--Jordan$\\big)$ algebras. We describe free objects in several $\\mathfrak{Ma}$-varieties. On the operadic level, over a field of characteristic $\\mathrm{char}\\,\\Bbbk\\neq 2$, we show $(\\mathfrak{Ma}\\text{-}\\Var)^{(!)}=\\mathfrak{Ma}\\text{-}(\\Var^{(!)})$ and that the Dong property is preserved by $\\mathfrak{Ma}.$ Finally, we obtain the necessary and sufficient condition for $\\mathfrak{Ma}\\text{-}\\Var=\\Var$. The second part of the paper is dedicated to another type of anti-algebras, called $\\mathcal{M}anin$ anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known $\\mathcal{M}anin$ anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a $\\mathcal{M}a$-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12479",
  "author_total": 3,
  "authors": [
    {
      "name": "Ivan Kaygorodov"
    },
    {
      "name": "Abror Khudoyberdiyev"
    },
    {
      "name": "Bauyrzhan Sartayev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5v5km2pxhkv8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12479",
  "fulltext_url": null,
  "id": "5v5km2pxhkv8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12479",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7221551568",
  "pdf_url": "https://arxiv.org/pdf/2610.12479",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5v5km2pxhkv8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12479",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12479"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A unified approach to the definition of anti-algebras",
  "title_text": "A unified approach to the definition of anti-algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/5v5km2pxhkv8f",
  "venue": "arXiv",
  "versions": []
}