<abstract xmlns="http://www.w3.org/1999/xhtml"><p>For a given integer 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0028_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo>=</mo><msubsup><mi>p</mi><mn>1</mn><mrow><msub><mi>α</mi><mn>1</mn></msub></mrow></msubsup><msubsup><mi>p</mi><mn>2</mn><mrow><msub><mi>α</mi><mn>2</mn></msub></mrow></msubsup><mo>⋯</mo><msubsup><mi>p</mi><mi>k</mi><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow></msubsup></math><tex-math>$n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$</tex-math></alternatives></inline-formula>
 (<italic>k</italic> ≥ 2), we give here a class of finitely presented finite monoids of order <italic>n</italic>. Indeed the monoids <italic>Mon</italic>(π), where
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0028_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mo>π</mo><mo>=</mo><mrow><mo stretchy="false">〈</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo stretchy="false">|</mo><msubsup><mi>a</mi><mi>i</mi><mrow><msubsup><mi>p</mi><mi>i</mi><mrow><msub><mi>α</mi><mi>i</mi></msub></mrow></msubsup></mrow></msubsup><mo>=</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><mo> </mo><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>a</mi><mi>i</mi></msub><msub><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><mo> </mo><mrow><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">〉</mo></mrow><mo>.</mo></math><tex-math>$$\pi = {\langle {a_1 ,a_2 , \ldots ,a_k |a_i^{p_i^{\alpha _i } } = {a_i}, {\left({i = 1,2, \ldots ,k} \right)}, a_i a_{i + 1} = {a_i}, \left({i = 1,2, \ldots ,k - 1} \right)} \rangle} .$$</tex-math></alternatives></disp-formula>
As a result of this study we are able to classify a wide family of the <italic>k-</italic>generated <italic>p</italic>-monoids (finite monoids of order a power of a prime <italic>p</italic>). An interesting di erence between the center of finite <italic>p</italic>-groups and the center of finite <italic>p</italic>-monoids has been achieved as well. All of these monoids are regular and non-commutative.</p></abstract>

For a given integer 
n=p1α1p2α2⋯pkαk$n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$
 (k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where
π=〈a1,a2,…,ak|aipiαi=ai, (i=1,2,…,k),aiai+1=ai, (i=1,2,…,k−1)〉.$$\pi = {\langle {a_1 ,a_2 , \ldots ,a_k |a_i^{p_i^{\alpha _i } } = {a_i}, {\left({i = 1,2, \ldots ,k} \right)}, a_i a_{i + 1} = {a_i}, \left({i = 1,2, \ldots ,k - 1} \right)} \rangle} .$$
As a result of this study we are able to classify a wide family of the k-generated p-monoids (finite monoids of order a power of a prime p). An interesting di erence between the center of finite p-groups and the center of finite p-monoids has been achieved as well. All of these monoids are regular and non-commutative.

Non-commutative finite monoids of a given order <italic>n</italic> ≥ 4

Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland, UK.

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, P.O. Box 14515/1775, Iran.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Non-commutative finite monoids of a given order  ≥ 4

For a given integer 
n=p1α1p2α2⋯pkαk$n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$
 (k ≥ 2), we give here a class of finitely presented finite...

Non-commutative finite monoids of a given order n ≥ 4

Published Online: Oct 20, 2015

Page range: 29 - 36

Received: Dec 01, 2012

Accepted: May 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0028

Keywords
Semigroup and monoid presentation, Regular semigroup, Inverse semigroup

© 2014 B. Ahmadi et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Non-commutative finite monoids of a given order n ≥ 4

Published Online: Oct 20, 2015

Page range: 29 - 36

Received: Dec 01, 2012

Accepted: May 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0028

KeywordsSemigroup and monoid presentation, Regular semigroup, Inverse semigroup

© 2014 B. Ahmadi et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Semigroup and monoid presentation, Regular semigroup, Inverse semigroup