It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B2|3(φ, ξ, ψ). In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number N ≥ 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N. For the numbers N = 6, 7, 8, 11, there do not exist singular N -dimensional superalgebras with a switch.
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We are grateful to the referee who carefully read the manuscript of the paper and made a series of useful comments.
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Translated from Algebra i Logika, Vol. 61, No. 6, pp. 742-765, November-December, 2022. Russian DOI:https://doi.org/10.33048/alglog.2022.61.605.
S. V. Pchelintsev and O. V. Shashkov are supported by Russian Science Foundation, grant No. 22-11-00081.
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Pchelintsev, S.V., Shashkov, O.V. Structure of Singular Superalgebras with 2-Dimensional Even Part and New Examples of Singular Superalgebras. Algebra Logic 61, 506–523 (2023). https://doi.org/10.1007/s10469-023-09716-z
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DOI: https://doi.org/10.1007/s10469-023-09716-z