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Nov
21
comment Set theory like first-order theory of ordered pairs
@AsafKaragila The atom axiom just says that a pair is equal to its left part if and only if it is equal to its right part. But the logic behind this axiom starts with the question what should be the left and right part of an atom. If it were the empty set, we would run into trouble with the axiom of extensionality. Hence the most useful definition seems to be that the left and right part of an atom are identical to the atom itself. Now we just need an axiom that allows us to turn this property into a convenient definition of what it means for a pair to be an atom.
Nov
21
revised Set theory like first-order theory of ordered pairs
added axiom scheme of induction and modified question accordingly
Nov
21
revised Set theory like first-order theory of ordered pairs
converted axioms to be sentences (no free variables) and improved formulation of axiom of pairing
Nov
21
comment Set theory like first-order theory of ordered pairs
@PeterSmith Both theories are intended as single-sorted theories. The intention of the first theory was to model a pair as a "universal data structure" in a programming language. The intention of the second theory was to avoid meaningless theorems like us.metamath.org/mpegif/avril1.html caused by definitions like the ordered pairs which feel like "hacks". But all two-sorted theories (including pairs in set theory) I tried felt even more like "hacks" than ordinary ZF set theory. But the theory of s-pairs doesn't feel worse than ordinary ZF to me, and seems to do the trick.
Nov
21
comment Set theory like first-order theory of ordered pairs
@AsafKaragila You are right, the axioms of ZF always use $\forall$ qualifiers for all free variables in the axioms, so I should have done the same. If it were possible, the theory should model "pairs" as they could be use as a "universal data structures" in a programming language. So there should be a finite number of atoms (I know how to achieve this), and each pair should be recursively defined as a pair of previously defined pairs or atoms (I believe that a first-order theory can't achieve this completely, and has to settle for something similar to PA).
Nov
21
comment Set theory like first-order theory of ordered pairs
@AsafKaragila These theories are intended to be single sorted, so $x$, $y$ in the axiom of pairing are pairs themselves (because every object of this theory is called a pair). I see that the textual description "If $x$ and $y$ are pairs..." is misleading, and will correct it.
Nov
21
asked Set theory like first-order theory of ordered pairs
Nov
2
comment Is the universal inverse semigroup of a commutative semigroup an embedding?
Reposted at mathoverflow. The question has been answered there with 'no', giving a counterexample based on the reference to "B. Schein described all semigroups embeddable into inverse semigroups in Schein, Boris M., Subsemigroups of inverse semigroups, Le Matematiche LI (1996), Supplemento, 205–227 (in fact the paper was written in the 50s)". I won't repost this as an answer here for the moment, because I have not worked through the reference yet, and so can't claim understand
Nov
2
accepted Relation between lattice theory and semilattice theory
Nov
1
answered Relation between lattice theory and semilattice theory
Nov
1
revised Relation between lattice theory and semilattice theory
Updated question based on what I learned in the meantime since I aksed this question
Oct
29
accepted Orders on the Cartesian product of partially ordered sets
Oct
28
answered Orders on the Cartesian product of partially ordered sets
Oct
28
answered Every poset is embedded into a meet-semilattice
Oct
28
comment What is the smallest variety of algebras containing all fields?
@Hurkyl Good observation that this a non-trivial (even so well known) fact. I tend to verify at mathematical structures whether I got such properties right. There is written "Classtype: variety" and the important identity (ensuring uniqueness) is "idempotents commute: $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$".
Oct
27
accepted What is the smallest variety of algebras containing all fields?
Oct
27
answered What is the smallest variety of algebras containing all fields?
Oct
26
comment Orders on the Cartesian product of partially ordered sets
I want to systematically describe these orders for posets, because I want to try to characterize those which give me bounded lattices (or semilattices with identity element). The application behind this is to better understand how an efficient data-strucure for a semilattice might look like (especially if I want to use a join-semilattice instead of a tree for representing hierarchies). Perhaps I would be better off thinking about relational representations instead, since this is what databases did when trees were no longer enough.
Oct
25
asked Orders on the Cartesian product of partially ordered sets
Oct
25
accepted Is the set of all definable subsets of the natural numbers recursively enumerable?