Flows, skew pairs and partial flows in regular semigroups

Suha Wazzan; David A. Oluyori; Suha Wazzan; David A. Oluyori

doi:10.3934/math.2026817

AIMS Mathematics

2026, Volume 11, Issue 7: 20103-20142. doi: 10.3934/math.2026817

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Flows, skew pairs and partial flows in regular semigroups

Suha Wazzan ^{1
,
,},
David A. Oluyori ²

1.
Department of Mathematics, Science Faculty, King Abdulaziz University, P.O. Box 42805, Jeddah 21589, Saudi Arabia
2.
Department of Applied Mathematics, School of Applied Sciences, Federal University of Technology, Babura, P.O. Box 2022, Jigawa State, Nigeria; daoluyori.mth@futb.edu.ng

Received: 26 April 2026 Revised: 29 June 2026 Accepted: 02 July 2026 Published: 09 July 2026
MSC : 20M17, 20M20, 18B40, 20M50

This work connected flows on regular semigroups with skew pairs of idempotents via the bipartite category $ \beta(S) $, where arrows $ e \xrightarrow{s} f $ exist if and only if some inverse $ s' $ satisfies $ e = ss' $ and $ f = s's $. Partial flows satisfying $ aba = a $ and $ bab = b $ were introduced. Using Luangchaisam and Changphas' theorem, explicit partial flows were constructed for strong, left regular, and right regular skew pairs, with domain sizes $ 1 $, $ 2 $ and $ 2 $, respectively. Flow monoids were classified for groups ($ \Phi(G) \cong G $), rectangular bands, Brandt semigroups ($ \Phi(B_n) \cong T_n $), inverse semigroups, and completely simple semigroups, where the sandwich matrix twists composition. The discrete skew pair case remains open.
- regular semigroups,
- flow monoids,
- skew pairs of idempotents,
- Cauchy categories,
- bipartite categories,
- partial transformation semigroups,
- Rees matrix semigroups,
- Brandt semigroups,
- inverse semigroups,
- categorical invariants
Citation: Suha Wazzan, David A. Oluyori. Flows, skew pairs and partial flows in regular semigroups[J]. AIMS Mathematics, 2026, 11(7): 20103-20142. doi: 10.3934/math.2026817

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Abstract

This work connected flows on regular semigroups with skew pairs of idempotents via the bipartite category $ \beta(S) $, where arrows $ e \xrightarrow{s} f $ exist if and only if some inverse $ s' $ satisfies $ e = ss' $ and $ f = s's $. Partial flows satisfying $ aba = a $ and $ bab = b $ were introduced. Using Luangchaisam and Changphas' theorem, explicit partial flows were constructed for strong, left regular, and right regular skew pairs, with domain sizes $ 1 $, $ 2 $ and $ 2 $, respectively. Flow monoids were classified for groups ($ \Phi(G) \cong G $), rectangular bands, Brandt semigroups ($ \Phi(B_n) \cong T_n $), inverse semigroups, and completely simple semigroups, where the sandwich matrix twists composition. The discrete skew pair case remains open.

References

[1]	A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, Mathematical Surveys, Vol. 7, American Mathematical Society, 1961.
[2]	J. M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, Oxford University Press, 1995.
[3]	N. D. Gilbert, Flows on regular semigroups, Appl. Categorical Struct., 11 (2003), 147–156. https://doi.org/10.1680/cien.156.4.147.36766 doi: 10.1680/cien.156.4.147.36766
[4]	S. A. Wazzan, Flows on classes of regular semigroups and Cauchy categories, J. Math., 2019 (2019), 8027391.
[5]	J. A. Conejero, V. Müller, A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342–348.
[6]	J. C. Oxtoby, S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. Math., 42 (1941), 874–920. https://doi.org/10.2307/1968772 doi: 10.2307/1968772
[7]	T. S. Blyth, M. H. Almeida Santos, Regular semigroups with skew pairs of idempotents, Semigroup Forum, 65 (2002), 264–274. https://doi.org/10.1007/s002330010112 doi: 10.1007/s002330010112
[8]	T. S. Blyth, M. H. Almeida Santos, Skew pairs of idempotents in transformation semigroups, Acta Math. Sinica, 22 (2006), 1705–1714. https://doi.org/10.1007/s10114-005-0703-5 doi: 10.1007/s10114-005-0703-5
[9]	T. Changphas, R. P. Sullivan, Skew idempotents in linear transformation semigroups, Linear Multilinear Algebra, 58 (2010), 399–411. https://doi.org/10.1080/03081080802443141 doi: 10.1080/03081080802443141
[10]	P. Luangchaisam, T. Changphas, Skew pairs of idempotents in partial transformation semigroups, Mathematics, 10 (2021), 2. https://doi.org/10.3390/math10010002 doi: 10.3390/math10010002
[11]	B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure Appl. Algebra, 48 (1987), 83–198. https://doi.org/10.1016/0022-4049(87)90108-3 doi: 10.1016/0022-4049(87)90108-3
[12]	A. Costa, B. Steinberg, The Schützenberger category of a semigroup, Semigroup Forum, 91 (2015), 543–559. https://doi.org/10.1007/s00233-014-9657-1 doi: 10.1007/s00233-014-9657-1
[13]	The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.11.0, 2020. Available from: https://www.gap-system.org.
[14]	F. Ates, E. G. Karpuz, C. Kocapinar, A. S. Cevik, Gröbner-Shirshov bases of some monoids, Discrete Math., 311 (2011), 1064–1071. https://doi.org/10.1016/j.disc.2011.03.008 doi: 10.1016/j.disc.2011.03.008
[15]	L. A. Bokut, Y. Chen, Y. Li, Gröbner-Shirshov bases for categories, In: C. Bai, L. Guo, J. L. Loday, Operads and universal algebra, World Scienfific, 2010, 1–23. https://doi.org/10.1142/9789814365123_0001
[16]	E. G. Karpuz, Gröbner-Shirshov bases of some semigroup constructions, Algebra Colloq., 22 (2015), 35–46. https://doi.org/10.1142/S100538671500005X doi: 10.1142/S100538671500005X
[17]	M. S. Putcha, Monoid algebras and their representations, Proceedings of the International Conference on Semigroups and Applications, 1996, 1–15.
[18]	B. Steinberg, Representation theory of finite monoids, Springer, 2016.

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