2017-10-09:
Notation (in the RPN-version):
- replaced f[U] by Uf_\exists throughout
- replaced f^{-1}[U] by Uf^{-1} throughout
- added end-of-example/definition/remark triangles
- in the power monoids: replaced "concatenation" by "multiplication"
p 05: new Proposition 2.01 stating that rational sets are closed under
direct images.
p 06: Proposition 0.2.04: renamed the finitely generated submonoid to M'
instead of S
p 07: dropped Remark 2.0.7; mentioned in the text that the inverse
image function \phi^{-1} and its right adjoint \phi_\forall need
not preserve multiplication; introduce lax and oplax homomorphisms
and show that \phi^{-1} is lax; unclear about \phi_\forall :-(
New Proposition 0.3.03: recognizable sets are stable under inverse images.
p 08: Replaced Corollary 0.3.05 by a Proposition stating that M-Rec a
Boolean subalgebra of MP.
p 08+: Definition 0.3.06 and Remark 0.3.07: used J resp. j for
pre-residuation and left quotients, reserved K and k for
post-residuations and right quotients; new equation (0.3-01)
p 10: Made the first part of the proof of Proposition 0.3.08 explicit.
p 11: new subsubsection on conter-examples; Rec is not closed
under direct images, composition and Kleene star.
p 13-14: lax and oplax functors are mentioned infomrally
p 31: alternative formulation of Vs_\exists and Vs_\forall
in terms of how cs^{-1} interacts with V; explicit formula number;
moved the terminology after the formula.
2017-12-16:
p 13: better intoduction to the notion of syntactic congruence
p 26: Example 2.1.08, second: added that grp is a full subcategory of mon
2018-01-23:
p 12: added the proof of Proposition 0.3.20.
p 13: completed the proof of Theorem 0.3.21.
p 17-21: revised and completed the section on M-automata
p 24: added an official definition of graph morphism and functor (4.0.01)
p 54: added a new section 4.8 on comma categories
p 55: added a new section 4.9 on profunctors (needed to properly deal
with ideals)
p 21: revised the section 3 on Green's Relations reflect the problems
encountered on Monday and fit with the categorial approach to
left/right/2-sided ideals in Section 4.9
2018-02-02:
p 31: new Remark 4.0.03 stating that the the hom-sets of the form
[C,C] are automatically monoids under composition, and that every
category may be collapsed into a (possibly large) pre-ordered set
by collapsing every non-empty hom-set to a singleton.
p 31: a new diagram illustrating the notion of "commutative
diagram"
2018-02-10:
p 29: started a new subsubsection on the proof of Schützenberger's
characterization of *-free languages utilizing Green's relations,
not yet completely typed up
p 52: Example 4.5.05: the result that spn is closed (item (1))
has been moved to Remarks 4.7.01 item (c) on p 59
p 60: new item (d) of Remark 4.7.01 to the effect that monads in
spn are just the small categories.
p 64: Example 4.9.02, first item: added a remark that the term
"order ideal" is slightly misleading in view of the later use of
the term "ideal"; for the latter domain and codomain have to
coincide, not so for order ideals.
p 64: new section 4.10 on ideals in monoids on a 2-category; this
subsumes the ideals introduced as special profunctors (moved here
from section 4.9), but also accounts for ring ideals (the
pertinent 2-category being the suspension of ab with tensor
product).
p 66+67: new Example 4.10.04 concerning ideals in monoids, rings and
categories. This is followed by some considerations on principal
ideals and a link between ideals and zero morphisms: collapsing
parallel morphisms in the former yields the latter, while the
former can be recovered as preimages of the latter.