On Faithful Matrix Representations of a Semilattice of Right-Zero Semigroups Whose Transformations Have Infinite Rank
DOI:
https://doi.org/10.32603/2071-2340-2025-4-15-26Keywords:
semigroup of idempotents, rank of a transformation, semilattice, faithful matrix representation of semigroupsAbstract
The paper examines semigroups of idempotents and the corresponding transformation semigroups, as well as the connection between the ranks of these transformations and the existence of a faithful matrix representation of the original semigroups. It is known that in the case where the transformations under consideration are right translations of elements of a semilattice of semigroups of right zeros, and these translations have finite rank, the semigroup has a faithful matrix representation. A conjecture was proposed that the finiteness of the rank of the transformations is a criterion for the existence of such a representation. However, the conjecture turned out to be false. The paper considers semilattices of two semigroups of right zeros in which the ranks of the transformations of elements of one of the semigroups are infinite. It is shown that among such semigroups there are those that have a faithful matrix representation and those that do not.
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Copyright (c) 2026 Larisa Zyablitseva, Svetlana Korabelshchikova, Elena Tolkacheva
This work is licensed under a Creative Commons Attribution 4.0 International License.
This work is licensed under a Creative Commons Attribution 4.0 International License.
