Let Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a real x such that every x–admissible ordinal is a cardinal in L. The known proofs of Harrington’s theorem “Det(Σ1) implies 0 ] exists” are done in two steps: first show that Det(Σ1) implies HP, and then show that HP implies 0] exists. The first step ...

We propose a generalization of non-commutative geometry and gauge theories based on ternary Z3-graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. These relations reflect the action of the Z3-group, which may be either trivial, i.e. abc = bca = cab, generalizing the usual commutativity, or non-tri...

We introduce Z3-graded objects which are the generalization of the more familiar Z2-graded objects that are used in supersymmetric theories and in many models of non-commutative geometry. First, we introduce the Z3graded Grassmann algebra, and we use this object to construct the Z3matrices, which are the generalizations of the supermatrices. Then, we generalize the concepts of supertrace and su...

The card game SET can be modeled by four-dimensional vectors over Z3. These vectors correspond to points in the affine four-space of order three (AG(4, 3)), where lines correspond to SETs, and in the affine plane of order nine (AG(2, 9)). SETless collections and other aspects of the game of SET will be explored through caps in AG(4, 3) and conics in AG(2, 9).

Many researchers have devoted themselves to the study of nowhere-zero flows and group connectivity. Recently, Thomassen confirmed the weak 3-flow conjecture, which was further improved by Lovász, Thomassen, Wu and Zhang who proved that every 6-edge-connected graph is Z3-connected. However, Conjectures 1 and 2 are still open. Conjecture 2 implies Conjecture 1 by a result of Kochol that reduces C...

Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifier-free BAPA (QFBAPA). In contrast to many other NP-complete problems (such as quantifier-free first-order logic or linear arithmetic), the appl...

We apply the standard approach of RG flow for the gauge couplings in N=1 D=4 Supergravity to show how to match its results with the heterotic Z3 orbifold and Type IIB Z3 orientifold-based models. Using only supergravity, anomaly cancellation and the requirement of unification we determine the part of the Kähler potential of the model invariant under the symmetries of the model. For heterotic or...