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=== Monoids ===
{{MOS|section|[[MOS:MATHSPECIAL]] for \ - needs {{tag|math}} markup|date=January 2024}}
A '''multiplicative monoid''' is a monoid ''A'' which also contains an [[absorbing element]] 0 (distinct from the identity 1 of the monoid), such that {{nowrap|1=0''a'' = 0}} for every ''a'' in the monoid ''A''. The field with one element is then defined to be {{nowrap|1='''F'''<sub>1</sub> = {{mset|0, 1}}}}, the multiplicative monoid of the field with two elements, which is [[initial object|initial]] in the category of multiplicative monoids. A '''monoid ideal''' in a monoid ''A'' is a subset ''I'' which is multiplicatively closed, contains 0, and such that {{nowrap|1=''IA'' = {{mset|''ra'' : ''r'' ∈ ''I'', ''a'' ∈ ''A''}} = ''I''}}. Such an ideal is '''prime''' if {{nowrap|''A'' ∖ ''I''}} is multiplicatively closed and contains 1.
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