Examples of Pomonoids of Full Transformations of a Poset

In this research, the partially ordered monoid (simple pomonoid) full transformations of a poset O(X) is studied, and some related properties are examined. We show that when the poset X_ is not totally ordered, the pomonoid of all decreasing singular self-maps of a poset X_ (denoted by S^-) and the pomonoid of all increasing singular self-maps of a poset X_ (denoted by S^+) may not be generally isomorphic. Some specific partial ordered relations are considered, and the cardinalities of S^- and S^+ under these relations are found. The set of fixed, decreasing, and increasing points of mapping α in O(X) are also investigated. KEYWORDS Posets, pomonoids, full transformations

Guaranteed Scoring Games

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by 'switching the players'. The order relation between two games is defined by comparing their stops in any disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

Monoid based semantics for linear formulas (corrected republication)

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

Monoid based semantics for linear formulas

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.