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. In this paper, necessary and sufficient conditions are studied for Bruck-Reilly and gener- alized Bruck-Reilly ∗-extensions of direct product of k monoids to be regular, unit regular, completely regular and orthodox. Moreover, we give an example of a finitely presented monoid (generalized Bruck-Reilly ∗-extension of Bruck-Reilly extension of a free group with infinite rank), the group of units of which is not finitely generated. 2010 Mathematics Subject Classification: 16S15; 20E06; 20E22. Keywords and Phrases: Generalized Bruck-Reilly ∗-extension, finite generation, finite presentability. Ozet. Bu makalede k tane monoidin direkt carpiminin Bruck-Reilly ve genellestirilmis Bruck-Reilly *- genislemelerinin, reguler, terslenebilir reguler, tamamen reguler ve orthodox olabilmesi icin gerek ve yeter kosullar calisilmistir.  Ayrica, biz terslenebilir elemanlarinin grubu sonlu uretecli olmayan sonlu sunumlu bir monoid (sonsuz rankli bir serbest grubun Bruck-Reilly genislemesinin genellestirilmis Bruck-Reilly *-genislemesi) ornegi verdik. Anahtar Kelimeler: Genellestirilmis Bruck-Reilly *-genislemesi, sonlu ureteclilik, sonlu sunumluluk.

Special Semigroup Classes Over Some Monoid Constructions and a New Example of a Finitely Presented Monoid with a Non-Finitely Generated Group of Units