Discussion Overview
The discussion revolves around the possibility of partitioning a semigroup into two subsemigroups, where one subsemigroup has an identity element and the other does not. The context includes theoretical considerations regarding semigroups, identity elements, and their properties.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions whether it is possible to partition a semigroup S into two subsemigroups S_1 and S_2, with S_1 having an identity element and S_2 having none.
- Another participant suggests that if the original semigroup has an identity, a specific partition can be made where S_1 consists of the identity element and S_2 contains all other elements.
- A later post clarifies that the original semigroup does not have an identity and asks if it is possible to create a partition where S_1 has a right-identity and S_2 has no identity element.
- One participant claims that if S_1 has a right-identity, then S_2 must have at least a left-identity, based on their proof involving specific element interactions.
Areas of Agreement / Disagreement
Participants express differing views on the conditions under which such a partition is possible, with no consensus reached on the validity of the claims made regarding identities in the subsemigroups.
Contextual Notes
The discussion includes assumptions about the properties of semigroups, particularly regarding identity elements and their implications for partitioning. The proof mentioned relies on specific observations that may not be universally applicable without further clarification.