Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .
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![\"\"](\"Edit_1798cf4c-b9a5-4ada-b2d4-9fbf535a6d28.png\")
be the partial symmetric semigroup on ![\"\"](\"Edit_86c31e58-0588-44fc-8ff3-78c73dc14be6.png\")
and let ![\"\"](\"Edit_09f7ec2a-49d1-463d-952e-93fbe00740cd.png\")
and ![\"\"](\"Edit_4aca07c1-9a36-44b6-83af-46fdd6278ca4.png\")
be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of ![\"\"](\"Edit_72d2bbdf-2b96-4812-a993-f49640bb98c9.png\")
, respectively. In this paper we investigate the cardinalities of ![\"\"](\"Edit_95038cdc-5496-495f-bfb9-29c817ba2df4.png\")
and ![\"\"](\"Edit_f8d3d782-ff8c-43b3-84af-d21a3f4291d5.png\")
, the set idempotents of ![\"\"](\"Edit_ac8237f8-83ec-4767-9029-b546377bf106.png\")
and ![\"\"](\"Edit_8b1bf8bf-7472-453c-965d-7308da5b4f4c.png\")
, respectively. We also investigate the cardinalities of certain equivalences on ![\"\"](\"Edit_e5f7aec4-67c1-48ba-8d03-ea6ceb2e7627.png\")
and ![\"\"](\"Edit_42a31943-fe9d-4ea8-8999-3aadb705fb38.png\")
.
