Some results on $$\mathcal {L}$$ L -commutative semigroups
- PDF / 1,810,081 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 113 Downloads / 166 Views
Some results on L‑commutative semigroups Roman S. Gigoń1 Received: 10 September 2019 / Accepted: 9 February 2020 © The Author(s) 2020
Abstract We prove first that every H-commutative semigroup is stable. Using this result [and some results from the standard text (Nagy, Special classes of semigroups, Kluwer, Dordrecht, 2001)], we give two equivalent conditions for a semigroup to be an archimedean H-commutative semigroup containing an idempotent element. It turns out that this result can be partially extended to L-commutative semigroups and quasicommutative semigroups. Keywords H-commutative semigroup · L-commutative semigroup · Quasicommutative semigroup · Archimedean semigroup · Stable semigroup · Nilextension
1 Introduction and preliminaries For undefined terminology of semigroup theory, the reader is referred to the books [3, 8, 9]. One of the most important decompositions in the theory of semigroups is a decomposition of a semigroup (induced by a semilattice congruence) into archimedean semigroups. In this field, one of the biggest contributions is due to Putcha, see [9, Chapter 2] (recall that a semigroup is a Putcha semigroup if and only if it is a semilattice of archimedean semigroups). Therefore, it is worth investigating certain classes of Putcha semigroups (especially, with idempotents), see, e.g., [9, Chapters 4–12]. On the other hand, many naturally arising semigroups are semilattice unions of archimedean semigroups. A good example is the semigroup of all n × n upper triangular matrices over a field, and more generally, the Zariski closure of any solvable linear algebraic group. Moreover, semilattices of archimedean semigroups Communicated by Victoria Gould. * Roman S. Gigoń [email protected] 1
Department of Mathematics, University of Bielsko-Biala, Ul. Willowa 2, 43‑309 Bielsko‑Biała, Poland
13
Vol.:(0123456789)
R. S. Gigoń
arise naturally in the theory of finite semigroups and languages, where the pseudovariety is denoted by 𝐃𝐒. Denote the set of idempotents of a semigroup S by ES , and the set of its regular elements by Reg(S). Let S be a semigroup and let A ⊆ S . Put √ A = {s ∈ S ∶ (∃ n ∈ ℤ+ )(sn ∈ A)}. Also, for each a ∈ S , let us put as usual: J(a) = S1 aS1 . Recall that a semigroup S is archimedean if for every ideal I of S, we have √ S = I. Remark 1 Observe that if in addition ES ≠ ∅ , then ES is contained in each ideal of S. Hence Reg(S) is a subset of a kernel of S (by a kernel of a semigroup A we mean (if it exists) the least ideal of A; this ideal is then denoted by KA). We say that a congruence 𝜌 on a semigroup S is a semilattice congruence if the semigroup S∕𝜌 is a semilattice. It is clear that the least semilattice congruence on an arbitrary semigroup always exists. Denote it by 𝜂 . It is also clear (and well-known) that in any semigroup J ⊆ 𝜂. Let C be some fixed class of semigroups (call its elements C-semigroups). A semigroup S is said to be a semilattice of C-semigroups if there exists a semilattice congruence 𝜌 on S such that every 𝜌-class of S is a C-semigroup (fo
Data Loading...
