Projection-primitive \(P\)-Ehresmann semigroups. (English) Zbl 1484.20103
Summary: \( P \)-Ehresmann semigroups introduced by Jones are natural generalizations of Ehresmann semigroups. The aim of this paper is to introduce and describe projection-primitive \(P \)-Ehresmann semigroups. More specifically, it is proved that a projection-primitive \(P \)-Ehresmann semigroup is either isomorphic to a Rees matrix semigroup over a monoid with some natural conditions or a generalized category with zero adjoined.
MSC:
| 20M10 | General structure theory for semigroups |
| 20M50 | Connections of semigroups with homological algebra and category theory |
References:
| [1] | K, Free locally inverse \(^\ast \)-semigroup, Czech. Math. J., 43, 523-545 (1993) · Zbl 0796.20044 · doi:10.21136/CMJ.1993.128418 |
| [2] | J, Abundant semigroups, Proc. Lond. Math. Soc., s3-44, 103-129 (1982) · Zbl 0481.20036 · doi:10.1112/plms/s3-44.1.103 |
| [3] | V. Gould, <i>Restriction and Ehresmann semigroups</i>, In: <i>Proceedings of the International Conference on Algebra 2010, Advances in Algebraic Structures</i>, World Sci. Publ., Hackensack, (2012), 265-288. · Zbl 1264.20067 |
| [4] | V. Gould, Notes on restriction semigroups and related structures, 2010. Available from: <a href=“www-users.york.ac.uk/varg1/restriction.pdf.” target=“_blank”>www-users.york.ac.uk/ varg1/restriction.pdf.</a> |
| [5] | J. M. Howie, <i>An introduction to semigroup theory</i>, London: Academic Press, 1976. · Zbl 0355.20056 |
| [6] | C, From right PP monoids to restriction semigroups: A survey, Eur. J. Pure Appl. Math., 2, 21-57 (2009) · Zbl 1214.20056 |
| [7] | P, A common framework for restriction semigroups and regular \(^\ast \)-semigroups, J. Pure Appl. Algebra, 216, 618-632 (2012) · Zbl 1257.20058 · doi:10.1016/j.jpaa.2011.07.014 |
| [8] | P, The semigroups \(B_2\) and \(B_0\) are inherently nonfinitely based, as restriction semigroups, Int. J. Algebra Comput., 23, 1289-1335 (2013) · Zbl 1293.20057 · doi:10.1142/S0218196713500264 |
| [9] | P, Varieties of \(P\)-restriction semigroups, Commun. Algebra, 42, 1811-1834 (2014) · Zbl 1303.20062 · doi:10.1080/00927872.2012.749883 |
| [10] | P, Varieties of restriction semigroups and varieties of categories, Commun. Algebra, 45, 1037-1056 (2017) · Zbl 1364.20040 · doi:10.1080/00927872.2016.1175457 |
| [11] | G, Partial monoid actions and a class of restriction semigroups, J. Algebra, 429, 342-370 (2015) · Zbl 1321.20052 · doi:10.1016/j.jalgebra.2015.01.017 |
| [12] | M. V. Lawson, <i>Inverse semigroups</i>, Singapore: World Scientific, 1998. · Zbl 1079.20505 |
| [13] | M, Semigroups and ordered categories. I. The reduced case, J. Algebra, 141, 422-462 (1991) · Zbl 0747.18007 · doi:10.1016/0021-8693(91)90242-Z |
| [14] | M, Rees matrix semigroups, Proc. Edinburgh Math. Soc., 33, 23-37 (1990) · Zbl 0668.20049 · doi:10.1017/S0013091500028856 |
| [15] | T, Regular \(^\ast \)-semigroups, Semigroup Forum, 16, 369-377 (1978) · Zbl 0408.20043 · doi:10.1007/BF02194636 |
| [16] | G, Inverse semigroups with minimal right ideals, J. London Math. Soc., 29, 404-411 (1954) · Zbl 0056.01904 |
| [17] | G, Matrix representations of inverse semigroups, J. Aust. Math. Soc., 9, 29-61 (1969) · Zbl 0185.04701 · doi:10.1017/S1446788700005656 |
| [18] | M. Petrich, <i>Inverse semigroups</i>, New York: John Wiley & Sons, Inc., 1984. · Zbl 0546.20053 |
| [19] | M, Certain varieties of completely regular \(^\ast \)-semigroups, Boll. Un. Mat. Ital. (B), 4, 343-370 (1985) · Zbl 0582.20039 |
| [20] | M, Embedding into almost left factorizable restriction semigroups, Commun. Algebra, 41, 1458-1483 (2013) · Zbl 1275.20064 · doi:10.1080/00927872.2011.643839 |
| [21] | P, Primitive orthodox semigroups, Semigroup Forum, 17, 365-372 (1979) · Zbl 0437.20049 · doi:10.1007/BF02194335 |
| [22] | Y, Primitive weakly \(B\)-orthodox semigroups, Southeast Asian Bull. Math., 36, 903-916 (2012) · Zbl 1289.20089 |
| [23] | S, Fundamental regular semigroups with quasi-ideal regular \(^\ast \)-transversals, Bull. Malays. Math. Sci. Soc., 38, 1067-1083 (2015) · Zbl 1331.20073 · doi:10.1007/s40840-014-0070-4 |
| [24] | S, On algebras of \(P\)-Ehresmann semigroups and their associate partial semigroups, Semigroup Forum, 95, 569-588 (2017) · Zbl 1486.20078 · doi:10.1007/s00233-017-9903-4 |
| [25] | S, An Ehresmann-Schein-Nambooripad-type theorem for a class of \(P\)-restriction semigroups, Bull. Malays. Math. Sci. Soc., 42, 535-568 (2019) · Zbl 1474.20113 · doi:10.1007/s40840-017-0497-5 |
| [26] | S, An Ehresmann-Schein-Nambooripad theorem for locally Ehresmann \(P\)-Ehresmann semigroups, Period. Math. Hung., 80, 108-137 (2020) · Zbl 1463.20067 · doi:10.1007/s10998-019-00309-x |
| [27] | P, Completions of generalized restriction \(P\)-restriction semigroups, Bull. Malays. Math. Sci. Soc., 43, 3651-3673 (2020) · Zbl 1509.20089 · doi:10.1007/s40840-020-00888-w |
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