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      On representing semigroups with subsemilattices

      Jackson, Marcel; Stokes, Tim E.
      DOI
       10.1016/j.jalgebra.2012.11.029
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      Jackson, M., & Stokes, T. (2013). On representing semigroups with subsemilattices. Journal of Algebra, 376, 228-260.
      Permanent Research Commons link: https://hdl.handle.net/10289/7232
      Abstract
      We examine the problem of representing semigroups as binary relations, partial maps and injective functions, with the constraint that certain pre-designated idempotent elements must be represented as restrictions of the identity function. Appropriately formulated, the corresponding classes of representable structures is a quasivariety, but we show that they cannot be finitely axiomatised in first order logic. Quite a few algebraic structures have both a semigroup reduct and the ability to distinguish certain idempotent elements, and we use our construction to show that representability for these is also not finitely axiomatisable. Amongst the classes covered are subsemigroups of weakly left ample semigroups, various classes of ordered semigroups, semigroups of various kinds of binary relation with fixset operation. We also give new proofs of the nonfinite axiomatisability of the class of semigroups of binary relations endowed with domain and/or range operations, and of subsemigroups of inverse semigroups.
      Date
      2013
      Type
      Journal Article
      Publisher
      Elsevier
      Collections
      • Computing and Mathematical Sciences Papers [1452]
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