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Question 1: Is there a classification of finite ordered semigroups (or monoids) with $n$ elements for small $n$? Is there computer algebra software that can generate those algebraic objects?

(Here I mean ordered in the sense of https://en.wikipedia.org/wiki/Ordered_semigroup )

Question 2: Is there a classification of finite ordered semigroups (or monoids) whose underlying poset $P$ is fixed? For example $P$ being a totally ordered chain or a Boolean lattice might be interesting.

Question 3: Is there such a classification as in question 2 at least for the much smaller class of inverse semigroups (or monoids) with their natural order? For example can one classify when the underlying poset is connected or a chain in a nice way or give even an explicit classification in the chain case?

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    $\begingroup$ Maybe you should make clear what you mean by ordered since there are a number of competing notions. Do you mean a non-strict partial order which is both left and right compatible? Anyway unordered semigroups are presumably ordered and they can't be classified. $\endgroup$ Commented Aug 15 at 1:10
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    $\begingroup$ You might look at Mark Sapir’s answer mathoverflow.net/q/93230. Classifying commutative linearly ordered monoids and the commutative 3-nilpotent ones are equally hard, as is classifying linearly ordered commutative magmas. These should all be really hard. Basically 3-nilpotent semigroups are unclassifiable. $\endgroup$ Commented Aug 15 at 3:41
  • $\begingroup$ @BenjaminSteinberg Thank you for the comments. I clarified that it is a compatible order and added a link at the end of the question. It seems question 2 is too hard so I added question 3 which specialises to inverse monoids. $\endgroup$ Commented Aug 15 at 22:01
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    $\begingroup$ For finite inverse semigroups the order is connected if and only if it has an absorbing element. It is a chain iff it is isomorphic to a chain of idempotents. $\endgroup$ Commented Aug 16 at 0:29

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