Mathematics

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition. In this note we attempt an introductory investigation of totally (also partially) ordered monoids, not assumed commutative (still writing them additively), and taking them as positive, i.e.\ every element is greater than the unit element. That, in the usual commutative cases, allows the ordering to be defined via the algebraic structure, namely, as divisibility (in our additive sense): a≤b defined as ∃c(b=a+c) . The noncommutative case offers several ways to generalize that. First we try to follow the divisibility definition (on the right or on the left). Then, alternatively, we insist on the ordering being compatible with the operation both on the left and on the right, but strict inequality may not carry over -- again refer to the ordinals example. We try to see what axiom(s) such requirements impose on the monoid structure, and some facts are established. Focusing especially on the totally ordered case, one finds that necessarily the noncommutativity is somewhat limited. One may partly emulate here the commutative case, speaking about infinitely grater vs.\ Archimedean to each other elements, and in the Archimedean case even emulate Euclid's Elements' theory of `ratios' -- all that imposing some partial commutativity.

We introduce the notion of (half) 2-adjoint equivalences in Homotopy Type Theory and prove their expected properties. We formalized these results in the Lean Theorem Prover.

We present a category equivalent to that of semi-Nelson algebras. The objects in this category are pairs consisting of a semi-Heyting algebra and one of its filters. The filters must contain all the dense elements of the semi-Heyting algebra and satisfy an additional technical condition. We also show that in the case of dually hemimorphic semi-Nelson algebras, the filters are not necessary and the category is equivalent to that of dually hemimorphic semi-Heyting algebras.

We show that under the proper forcing axiom the class of all Aronszajn lines behave like σ -scattered orders under the embeddability relation. In particular, we are able to show that the class of better quasi order labeled fragmented Aronszajn lines is itself a better quasi order. Moreover, we show that every better quasi order labeled Aronszajn line can be expressed as a finite sum of labeled types which are algebraically indecomposable. By encoding lines with finite labeled trees, we are also able to deduce a decomposition result, that for every Aronszajn line L there is integer n such that for any finite colouring of L there is subset L ′ of L isomorphic to L which uses no more than n colours.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind's axiomatization of the natural number system. The latter is based on a structure (N,0,s) consisting of a set N , a distinguished element 0∈N and a function s:N→N . The structure in our axiomatization is a triple (O,L,s) , where O is a class, L is a function defined on all s -closed `subsets' of O , and s is a class function s:O→O . In fact, we develop the theory relative to a Grothendieck-style universe (minus the power-set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.

There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the "existence'" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arbitrary level of precision, or (iii) our ability, "in theory", to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the "continuum" of real numbers, which extends in the plane to the complex numbers and gives us, among other things, "rigorous" definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra -- the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over C has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of "existence" and "in theory'" in (i) to (iii) above.

We describe a formal proof of the independence of the continuum hypothesis ( CH ) in the Lean theorem prover. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of ¬CH and a ? -closed forcing for the consistency of CH .

An operator set is functionally incomplete if it can not represent the full set {¬,∨,∧,→,↔} . The verification for the functional incompleteness highly relies on constructive proofs. The judgement with a large untested operator set can be inefficient. Given with a mass of potential operators proposed in various logic systems, a general verification method for their functional completeness is demanded. This paper offers an universal verification for the functional completeness. Firstly, we propose two abstract operators R ˆ and R ˘ , both of which have no fixed form and are only defined by several weak constraints. Specially, R ˆ ≥ and R ˘ ≥ are the abstract operators defined with the total order relation ≥ . Then, we prove that any operator set R is functionally complete if and only if it can represent the composite operator R ˆ ≥ ∘ R ˘ ≥ or R ˘ ≥ ∘ R ˆ ≥ . Otherwise R is determined to be functionally incomplete. This theory can be generally applied to any untested operator set to determine whether it is functionally complete.

The following paper presents a heuristic method by which sum-of-product Boolean expressions can be simplified with a specific focus on the removal of redundant and selective prime implicants. Existing methods, such as the Karnaugh map and the Quine-McCluskey method [1, 2], fail to scale since they increase exponentially in complexity as the quantity of literals increases, doing as such to ensure the solution is algorithmically obtained. By employing a heuristic model, nearly all expressions can be simplified at an overall reduction in computational complexity. This new method was derived from the fundamental Boolean laws, Karnaugh mapping, as well as truth tables.

The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of structures coincide with isomorphisms. We prove a version of this principle for a wide range of higher-categorical structures, adapting FOLDS-signatures to specify a general class of structures, and using two-level type theory to treat all categorical dimensions uniformly. As in the previously known case of 1-categories (which is an instance of our theory), the structures themselves must satisfy a local univalence principle, stating that identifications coincide with "isomorphisms" between elements of the structure. Our main technical achievement is a definition of such isomorphisms, which we call "indiscernibilities", using only the dependency structure rather than any notion of composition.