symmetric monoidal (∞,1)-category of spectra
… idempotent semigroup …
Recall that a band is a semigroup in which every element is idempotent.
Commutative bands are usually known as semilattices. This is the semigroup-theoretic definition, but there is also an order theoretic definition: given a semilattice $L$ in this semigroup-theoretic sense, it has a canonical partial order given by $e\precreq f$ when $e f = e$. So semilattices are also posets.
Finitely generated bands are finite: see Howie 76, Section IV.4.
Now a rectangular band may be described as a semigroup satisfying the identity $a b a = a$ for all elements $a$ and $b$.
A rectangular band is indeed a band since the defining identity implies $x y z = x z$ for all $x,y,z$ whence by taking $y=z=x$ one gets $x x x = x x$ and from the defining identity $x x x=x$ hence $x x = x$. In order to get the first equation expand $x y z$ by substituting $x z x$ for $x$: $x y z = (x z x) y z = x (z (xy) z) = x z \; .$
If $S$ is a rectangular band, then there exist non-empty sets $I$ and $J$ such that $S$ is isomorphic as a semigroup to $I\times J$ equipped with the multiplication $(i, j)(p,q) = (i,q)$ for $i,p\in I$ and $j,q\in J$.
Every band $S$ has a decomposition as a disjoint union $\coprod_{x\in L} R_x$ where $L$ is a semilattice, each $R_x$ is a sub-semigroup that is a rectangular band, and $R_x R_y \subseteq R_{x y}$ for every $x$ and $y$. This is a bit weaker than saying we have a functor from the poset $L$ to the category of rectangular bands, because we lack connecting morphisms $R_x \to R_y$.
Let $Rect$ be the category of rectangular bands with semigroup homomorphisms as morphisms.
Since rectangular bands are an equationally defined subclass of the class of all semigroups, $Rect$ is a subvariety of the variety $SGr$ of semigroups and hence enjoys all the usual (co)completeness properties of a variety.
Since $Rect$ is the 2-valued collapse? of the topos $Set\times Set$ it is even a cartesian closed variety. Since the distributive law holds for (finite) coproducts in cartesian closed categories, $Rect$ is a distributive category. Since it is not locally cartesian closed it is neither a topos nor even an extensive category. For more on this see Johnstone (1990).
collapsed topos?
J. Howie, An introduction to semigroup theory, Academic Press 1976.
K. S. Brown, Semigroups, Semirings, and Markov Chains, J. Theor. Prob. 13 no.3 (2000) pp.871-938. (arXiv:math/0006145)
Peter Johnstone, Collapsed toposes and cartesian closed varieties , JPAA 129 (1990) pp.446-480.
N. Kimura, The structure of idempotent semigroups I , Pacific Journal of Mathematics 8 no.2 (1958) pp.257-275. (pdf)
Stuart Margolis, Franco Saliola, Benjamin Steinberg, Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry (arXiv:1508.05446)
Last revised on January 9, 2021 at 10:44:54. See the history of this page for a list of all contributions to it.