Poly-infix operators and operator families are introduced as an alternative for working modulo associativity and the corresponding bracket deletion convention. Poly-infix operators represent the basic intuition of repetitively connecting an ordered s
equence of entities with the same connecting primitive.
Proposition algebra is based on Hoares conditional connective, which is a ternary connective comparable to if-then-else and used in the setting of propositional logic. Conditional statements are provided with a simple semantics that is based on evalu
ation trees and that characterizes so-called free valuation congruence: two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. Free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. Valuation congruences that identify more conditional statements than free valuation congruence are repetition-proof, contractive, memorizing, and static valuation congruence. Each of these valuation congruences is characterized using a transformation on evaluation trees: two conditional statements are C-valuation congruent if, and only if, their C-transformed evaluation trees are equal. These transformations are simple and natural, and only for static valuation congruence a slightly more complex transformation is used. Also, each of these valuation congruences is axiomatized in proposition algebra. A spin-off of our approach can be called normalization functions for proposition algebra: for each valuation congruence C considered, two conditional statements are C-valuation congruent if, and only if, the C-normalization function returns equal images.
In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce fracpairs as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zer
o. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a common meadow, which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element a that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering a as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.
Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with a that propagates through all operations of the meadow signature (this additional value can be interpreted as an error elemen
t). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero that admit a certain cancellation law.
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.
Let Q_0 denote the rational numbers expanded to a meadow, that is, after taking its zero-totalized form (0^{-1}=0) as the preferred interpretation. In this paper we consider cancellation meadows, i.e., meadows without proper zero divisors, such as $Q
_0$ and prove a generic completeness result. We apply this result to cancellation meadows expanded with differentiation operators, the sign function, and with floor, ceiling and a signed variant of the square root, respectively. We give an equational axiomatization of these operators and thus obtain a finite basis for various expanded cancellation meadows.
A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and so-called
repeat instructions, a particular type of control instructions. The resulting mathematical structure is a semigroup. We relate this set of programs to program algebra (PGA) and show that a particular subsemigroup is a carrier for PGA by providing axioms for single-pass congruence, structural congruence, and thread extraction. This subsemigroup characterizes periodic single-pass instruction sequences and provides a direct basis for PGAs toolset.
A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatiza
tion of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a differential cancellation meadow.
Analytic execution architectures have been proposed by the same authors as a means to conceptualize the cooperation between heterogeneous collectives of components such as programs, threads, states and services. Interface groups have been proposed as
a means to formalize interface information concerning analytic execution architectures. These concepts are adapted to organization architectures with a focus on financial transfers. Interface groups (and monoids) now provide a technique to combine interface elements into interfaces with the flexibility to distinguish between directions of flow dependent on entity naming. The main principle exploiting interface groups is that when composing a closed system of a collection of interacting components, the sum of their interfaces must vanish in the interface group modulo reflection. This certainly matters for financial transfer interfaces. As an example of this, we specify an interface group and within it some specific interfaces concerning the financial transfer architecture for a part of our local academic organization. Financial transfer interface groups arise as a special case of more general service architecture interfaces.
