Finite Linearly Ordered Abelian Monoids

Question

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids".

The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The sequence is also described as "semi-groups with the greatest element of the corresponding chain as neutral element".

I don't understand why the neutral element (identity) needs to be the largest of the chain. This really leads to two issues for me.

1. I don't see why a linearly ordered semigroup with identity needs to have the identity as the largest element. Although I could understand why those where the identity is the largest are identified with those where the identity is the smallest (by reversing the order). These aren't order isomorphic, but I could see the reason to identify them.

2. The real problem for me is considering ordered monoids where the identity falls somewhere else in the order.

Here is an example of what I am talking about.

We want to count the number of linearly ordered abelian monoids of order $4$. The sequence A030453 says this number is $6$. When assuming the identity is maximal, this makes sense. Here are the matrix representations for the $6$ monoids. I am assuming the nonzero elements of the monoid are $\{a,b,c\}$, with $c<b<a<0$. I am only including nontrivial additions.

$ \begin{array}{c| c c c} & a & b & c \\\hline a & b & c & c \\ b & c & c & c \\ c & c & c & c \end{array} $ $ \begin{array}{c| c c c} & a & b & c \\\hline a & a & b & c \\ b & b & c & c \\ c & c & c & c \end{array} $ $ \begin{array}{c| c c c} & a & b & c \\\hline a & c & c & c \\ b & c & c & c \\ c & c & c & c \end{array} $ $ \begin{array}{c| c c c} & a & b & c \\\hline a & b & b & c \\ b & b & b & c \\ c & c & c & c \end{array} $ $ \begin{array}{c| c c c} & a & b & c \\\hline a & a & c & c \\ b & c & c & c \\ c & c & c & c \end{array} $ $ \begin{array}{c| c c c} & a & b & c \\\hline a & a & b & c \\ b & b & b & c \\ c & c & c & c \end{array} $

Next, I want to consider a monoid where the identity is properly within the chain. Say $c<b<0<a$, with multiplication table

$ \begin{array}{c| c c c} & a & b & c \\\hline a & a & a & a \\ b & a & c & c \\ c & a & c & c \end{array} $

So my questions are:

1. Is this array somehow considered the "same" as one of the six previous ones?

2. If the answer to the first question is no, then why is this not counted as a linearly ordered monoid of size $4$ in A030453?

3. More to the point, is there something in the definition of "finite linearly ordered abelian monoid" that forces the identity element to be maximal (or minimal) in the ordering?

Answer 1

This is more like a long comment; anyway, terminology in this area is far from standard.

If memory serves, Wehrung (just Google the name in connection with ordered monoids) defines that a positively ordered monoid is a commutative monoid equipped with a preorder such that the neutral element $0$ satisfies the axiom $0 \lesssim x,$ subject to the condition that $+$ is order-preserving in each argument. Hence OEIS is listing what Wehrung would call the linearly ordered POMs.

I do not know of any axiom that forces $0 \lesssim x$ except for $x \lesssim x+y$, which is clearly equivalent.