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In this paper we study submonoids of the monoid I ր

sn the pper we show tht the monoid IN∞ of ll prtil o(nite isometriE es of positive integers does not emed isomorphilly into the monoid ID∞ of ll prtil o(nite isometries of integersF woreoverD every nonEnnihilting homomorphism h : IN∞ → ID∞ hs the following propertyX the imge (IN∞)h is isomorphi either to the twoEelement yli group Z2 or to the dditive group of integers Z(+)F elso we prove tht the monoid IN∞ is not (nitely genertedD ndD moreoverD monoid IN∞ does not ontin miniml generting setF Key words: prtil isometryD inverse semigroupD prtil ijetionD iE yli monoidD isomorphi emeddingD group ongrueneD genertorD miniml generting setF 2020 Mathematics Subject Classication: 20M20, 20M30

Ìè êîðèñòóâàòèìåìîñü òåðìiíîëîãi¹þ ç [8, 16, 17].Íàäàëi ó òåêñòi ïîòóaeíiñòü ìíîaeèíè A ïîçíà÷àòèìåìî ÷åðåç |A| i ìíîaeèíó íàòóðàëüíèõ ÷èñåë ÷åðåç N.ßêùî âèçíà÷åíå ÷àñòêîâå âiäîáðàaeåííÿ α : X Y ç ìíîaeèíè X ó ìíîaeèíó Y , òî ÷åðåç dom α i ran α áóäåìî ïîçíà÷àòè éîãî îáëàñòü âèçíà÷åííÿ òà îáëàñòü çíà÷åíü, âiäïîâiäíî, à ÷åðåç (x)α i (A)α îáðàçè åëåìåíòà x ∈ dom α òà ïiäìíîaeèíè A ⊆ dom α ïðè ÷àñòêîâîìó âiäîáðàaeåííi α, âiäïîâiäíî. ×àñòêîâå âiäîáðàaeåííÿ α : X Y íàçèâà¹òüñÿ êî-ñêií÷åííèì, ÿêùî ìíîaeèíè X \ dom α òà Y \ ran α ¹ ñêií-÷åííèìè.ßêùî S íàïiâãðóïà, òî ¨¨ïiäìíîaeèíà iäåìïîòåíòiâ ïîçíà÷à¹òüñÿ ÷åðåç E(S). Íàïiâãðóïà S íàçèâà¹òüñÿ iíâåðñíîþ, ÿêùî äëÿ äîâiëüíîãî ¨¨åëåìåíòà x iñíó¹ ¹äèíèé åëåìåíò x −1 ∈ S òàêèé, ùî xx −1 x = x òà x −1 xx −1 = x −1 .  iíâåðñíié íàïiâãðóïi S âèùå îçíà÷åíèé åëåìåíò x −1 íàçèâà¹òüñÿ iíâåðñíèì äî x. Â'ÿçêà öå íàïiâãðóïà iäåìïîòåíòiâ, à íàïiâ ðàòêà öå êîìóòàòèâíà â'ÿçêà.Íåõàé I λ ìíîaeèíà âñiõ ÷àñòêîâèõ âçà¹ìíî îäíîçíà÷íèõ ïåðåòâîðåíü íåíóëüîâîãî êàðäèíàëà λ ç âèçíà÷åíîþ íà íié íàïiâãðóïîâîþ îïåðàöi¹þ x(αβ) = (xα)β ÿêùî x ∈ dom(αβ) = {y ∈ dom α : yα ∈ dom β}, äëÿ α, β ∈ I λ .

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