arXiv:2208.05237v1 [math.RA] 10 Aug 2022 COMMUTATORS IN REES MATRIX SEMIGROUPS JELENA RADOVI ´ C AND NEBOJ ˇ SA MUDRINSKI Jelena Radovi´ c 1 Department of Mathematics University of East Sarajevo 71123 East Sarajevo Bosnia and Herzegovina ORCID: 0000-0002-1023-4463 jelena.radovic@ff.ues.rs.ba Nebojˇ sa Mudrinski 2 Department of Mathematics and Informatics Faculty of Sciences University of Novi Sad 21000 Novi Sad Serbia ORCID: 0000-0001-9830-6603 nmudrinski@dmi.uns.ac.rs Abstract. We study the centralizing condition and commutators on Rees matrix semigroups. We obtain a complete characterization of the binary com- mutator on Rees matrix semigroups, and use it to study other properties of the commutator. Consequently, we deduce that a Rees matrix semigroup is nilpotent (solvable) if and only if its maximal subgroup is nilpotent (solvable). Keywords and phrases. completely simple semigroups, commutators, nilpo- tency, solvability 1. Introduction Let S =(S, ·) denote a semigroup, that is, an algebra with a associative binary operation · on a nonempty set S . We say that an algebra A satisfies the term condition, in abbreviation A is a TC algebra, if for every (n + 1)-ary term t = t(x, y), and for every a, b ∈ A, c, d ∈ A n we have t(a, c)= t(a, d) = ⇒ t(b, c)= Date : August 11, 2022. 2000 Mathematics Subject Classification. 08A30, 20M17. 1 Corresponding author 2 Supported by the Ministry of Science, Education and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022- 14/200125) 1