A Primer on Power Semigroups

Prof. Salvatore Tringali
Hebei Normal University, Shijiazhuang, China

Abstract

The course will consist of three two-hour lectures and is aimed at students and researchers with a basic background in algebra (in particular, familiarity with groups and semigroups at an introductory level). The overall goal is to survey recent developments in the theory of power semigroups and power monoids.

The large power semigroup of a semigroup S (written multiplicatively) is the semigroup P(S) of all non-empty subsets of S, endowed with the operation of setwise product naturally induced by S on its power set. If, in particular, S contains an identity element, then P(S) is a monoid, accordingly called the large power monoid of S. Power semigroups and power monoids lay more or less dormant for about two decades, but have attracted renewed interest in recent years.

Outline

In the first lecture, we will introduce power semigroups and power monoids and discuss various instances of a rather general problem (formulated in the language of categories and functors) for these structures and some related constructions. We will start with the TamuraShafer problem for power semigroups [9] (also in the nitary variant set forth in [8]), then move on to an isomorphism problem for reduced nitary power monoids stemming from a conjecture of Bienvenu and Geroldinger [2], and examine its solution in a number of important cases [13, 15, 5, 7]. Finally, if time permits, we will consider an isomorphism problem for monoids of ideals set forth in [4].

The second lecture will be devoted to the study of automorphisms of power semigroups and power monoids. We will discuss known results and open questions concerning the extent to which automorphism groups of power constructions reect the structure of the underlying semigroups and monoids. This part will essentially be a thorough review of [14, 11, 6, 16, 17].

In the third lecture, we will explore the arithmetic properties of power monoids [1, 3], with an emphasis on recent progress and open problems. The presentation will be largely based on the survey [10], and will touch on factorization-theoretic aspects.

Schedule

• 23/6/2026, 11:00-13:00
• 24/6/2026, 11:00-13:00
• 26/6/2026, 11:00-13:00

Exam
The final assessment (which will be overseen by Laura Cossu) consists of one of the following options: an in‑depth seminar on a topic covered during the lectures, or a written essay on selected course topics.

References

[1] A. A. Antoniou and S. Tringali, On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups, Pacific J. Math. 312 (2021), No. 2, 279-308.
[2] P.-Y. Bienvenu and A. Geroldinger, On algebraic properties of power monoids of numerical monoids, Israel J. Math. 265 (2025), 867-900.
[3] L. Cossu and S. Tringali, On the arithmetic of power monoids, J. Algebra 686 (Jan 2026), 793-813.
[4] P. A. García-Sánchez and S. Tringali, Semigroups of ideals and isomorphism problems, Proc. Amer. Math. Soc. 153 (2025), No. 6, 2323-2339 Draft
[5] B. Rago, The isomorphism problem for reduced finitary power monoids, preprint (arXiv:2601.22469).
[6] B. Rago, The automorphism group of reduced power monoids of finite abelian groups, preprint (arXiv:2510.17533).
[7] B. Rago, A counterexample to an isomorphism problem for power monoids, Proc. Amer. Math. Soc., to appear (arXiv:2509.23818).
[8] S. Tringali, On the isomorphism problem for power semigroups, pp. 429437 in: M. Brear, A. Geroldinger, B. Olberding, and D. Smertnig (eds.), Recent Progress in Ring and Factorization Theory (Graz, Austria, July 10-14, 2023), Springer Proc. Math. Stat. 477, Springer, 2025.
[9] T. Tamura and J. Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32; Errata, ibid. 29 (1984), No. 4, 679.
[10] S. Tringali, Power monoids and their arithmetic: a survey, Amer. Math. Monthly, to appear (https://arxiv.org/abs/2602.15754).
[11] S. Tringali and K. Wen, The Additive Group of Integers and the Automorphisms of its Power Monoids, preprint (arXiv:2504.12566).
[12] S. Tringali and K. Wen, On the automorphisms of the power semigroups of a numerical semigroup, preprint (not yet on arXiv).
[13] S. Tringali and W. Yan, A conjecture of Bienvenu and Geroldinger on power monoids, Proc. Amer. Math. Soc. 153 (2025), No. 3, 913-919.
[14] S. Tringali and W. Yan, On power monoids and their automorphisms, J. Combin. Theory Ser. A 209 (2025), 105961
[15] S. Tringali and W. Yan, Torsion groups and the BienvenuGeroldinger conjecture, Bull. London Math. Soc., to appear (arXiv:2601.19592).
[16] D. Wong, S. Xu, C. Zhang, and J. Zhao, On automorphism groups of power semigroups over numerical semigroups or over numerical monoids, preprint (arXiv:2512.12606).
[17] D. Wong, S. Xu, C. Zhang, and Z. Wang, On automorphism group of the reduced nitary power monoid of the additive group of integers, preprint (arXiv:2602.19493).

Acknowledgements

The course is partially supported by the Visiting Professors Program of the University of Cagliari.