Some categorical equivalences for -unitary inverse semigroups

Author:
Mario Petrich

Journal:
Trans. Amer. Math. Soc. **259** (1980), 493-503

MSC:
Primary 20M10; Secondary 18B10

MathSciNet review:
567092

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Abstract: The structure of *E*-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of *E*-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.

**[1]**D. B. McAlister,*Groups, semilattices and inverse semigroups. I, II*, Trans. Amer. Math. Soc.**192**(1974), 227–244; ibid. 196 (1974), 351–370. MR**0357660**, 10.1090/S0002-9947-1974-0357660-2**[2]**-,*Groups, semilattices and inverse semigroups*. II, Trans. Amer. Math. Soc.**196**(1974), 351-370.**[3]**Saunders MacLane,*Categories for the working mathematician*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR**0354798****[4]**W. D. Munn,*A note on 𝐸-unitary inverse semigroups*, Bull. London Math. Soc.**8**(1976), no. 1, 71–76. MR**0396809****[5]**N. R. Reilly and W. D. Munn,*𝐸-unitary congruences on inverse semigroups*, Glasgow Math. J.**17**(1976), no. 1, 57–75. MR**0404498****[6]**M. Petrich and N. R. Reilly,*A representation of 𝐸-unitary inverse semigroups*, Quart. J. Math. Oxford Ser. (2)**30**(1979), no. 119, 339–350. MR**545069**, 10.1093/qmath/30.3.339**[7]**Boris M. Schein,*A new proof for the McAlister “𝑃-theorem”*, Semigroup Forum**10**(1975), no. 2, 185–188. MR**0379716**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0567092-2

Keywords:
*E*-unitary inverse semigroups,
equivalence of categories,
McAlister triples,
unitary triples

Article copyright:
© Copyright 1980
American Mathematical Society