Structures and ∗-Embeddability of Commutative Semigroups with Involution of Order 3, 4, and 5
DOI:
https://doi.org/10.58987/6m7qh323Keywords:
Semigroup, ∗-Semigroup ring, ∗-Embeddability, InvolutionAbstract
This paper investigates the ∗-embeddability of proper ∗-semigroups into rings with proper involution. We establish a criterion characterizing ∗-embeddability in terms of proper ∗-ideals of the semigroup ring ℤ[S]. In particular, if the equation XX* = 0 has only the trivial solution in ℤ[S], then (S,*) is ∗-embeddable in ℤ[S]. We also give examples illustrating both ∗-embeddability and the failure of ∗-embeddability. Finally, we computationally investigate all commutative proper ∗-semigroups of orders 3, 4, and 5, consisting of 5072 cases in total. Computation shows that 4940 cases have only the trivial solution to XX* = 0, while 132 cases admit nontrivial solutions. In every nontrivial case, the computation finds a solution with coefficients in {-1,0,1} whose sum is 0, suggesting a general pattern for solutions of XX* = 0.
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