Representations of inverse semigroups in complete atomistic inverse meet-semigroups. (English) Zbl 1508.20110
Summary: As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in \(\mathscr{I}_X \), a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse \(\wedge\)-semigroups is developed. This class of inverse \(\wedge\)-semigroups, otherwise known as inverse algebras, includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.
Keywords:
inverse semigroups; representation of semigroups; inverse meet semigroups; partial automorphism monoidsReferences:
| [1] | Clifford, AH; Preston, GB, The Algebraic Theory of Semigroups. Mathematical Surveys 7 (1961), Providence: American Mathematical Society, Providence · Zbl 0111.03403 |
| [2] | Ellerman, D., The logic of partitions: introduction to the dual of the logic of subsets, Rev. Symb. Log., 3, 287-350 (2010) · Zbl 1211.03101 · doi:10.1017/S1755020310000018 |
| [3] | Ellerman, D., An introduction to partition logic, Log. J. IGPL, 22, 94-125 (2014) · Zbl 1343.03026 · doi:10.1093/jigpal/jzt036 |
| [4] | FitzGerald, DG; Leech, J., Dual symmetric inverse monoids and representation theory, J. Aust. Math. Soc. Ser. A, 64, 345-367 (1998) · Zbl 0927.20040 · doi:10.1017/S1446788700039227 |
| [5] | Lawson, MV, Inverse Semigroups: The Theory of Partial Symmetries (1998), Singapore: World Scientific, Singapore · Zbl 1079.20505 |
| [6] | Lawson, MV, Non-commutative Stone duality: inverse semigroups, topological groupoids and \(C^{\ast }\)-algebras, Int. J. Algebra Comput., 22, 6, 1250058 (2012) · Zbl 1273.22006 · doi:10.1142/S0218196712500580 |
| [7] | Lawson, MV; Margolis, SW; Steinberg, B., The étale groupoid of an inverse semigroup as a groupoid of filters, J. Aust. Math. Soc., 94, 234-256 (2013) · Zbl 1297.20067 · doi:10.1017/S144678871200050X |
| [8] | Leech, J., Inverse monoids with a natural semilattice ordering, Proc. Lond. Math. Soc. (3), 70, 146-182 (1995) · Zbl 0830.20086 · doi:10.1112/plms/s3-70.1.146 |
| [9] | Leech, J., On the foundations of inverse monoids and inverse algebras, Proc. Edinb. Math. Soc. (2), 41, 1-21 (1998) · Zbl 0909.20048 · doi:10.1017/S0013091500019398 |
| [10] | Petrich, M., Inverse Semigroups (1984), New York: Wiley, New York · Zbl 0546.20053 |
| [11] | Schein, BM, The minimal degree of noble inverse semigroups, Contrib. Gen. Algebra, 6, 247-252 (1988) · Zbl 0701.20039 |
| [12] | Schein, BM, The minimal degree of a finite inverse semigroup, Trans. Am. Math. Soc., 333, 2, 877-888 (1992) · Zbl 0770.20029 · doi:10.1090/S0002-9947-1992-1072101-9 |
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