Summand absorbing submodules of a module over a semiring

An R-module V over a semiring R lacks zero sums (LZS) if $x+y=0$ implies $x=y=0$. More generally, we call a submodule W of V “summand absorbing” (SA) in V if $?x,y\in V:x+y\in W?x\in W,y\in W$. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.     

Existentially Incomplete Tame Models and a Conjecture of Ellentuck

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II₂ arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some do not.     

Hölder and Minkowski type inequalities for pseudo-integral

There are proven generalizations of the Hölder’s and Minkowski’s inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup, ⊙), where ⊙ is generated and the third semiring where both pseudo-operations are idempotent, i.e., ⊕ = sup and ⊙ = inf.     

Semiring and Semimodule Issues in MV-Algebras

It is known that an atomic right LCM domain need not be a UFD but is a projectivity-UFD if it is also modular. This paper studies a slightly weaker and easier condition, the RAMP (acronym for the property in the title) , which also ensures that an atomic right LCM domain will be a projectivity-UFD. Among other things it is shown that in an atomic LCM domain, modularity is equivalent to the pair RAMP and LAMP (the left-right analog of RAMP). This result is then used to show that an atomic LCM domain with conjugation is modular. An example is given of an atomic LCM domain that has neither the RAMP nor the LAMP. All rings are not-necessarily commutative integral domains. Recall that an atomic ring is one in which every nonzero nonunit is a product of atoms (i.e. irreducibles) . A ring R is a right LCM domain if for any two elements a and b in R, aR ∩ bR is a principal right ideal. A right LCM domain need not be a left LCM domain [3] . If a ring has both properties it is called an LCM domain. It Is known (see Example 2 below) that, unlike the commutative case, an atomic right LCM domain need not be a UFD (unique factorization domain). In [1] it is shown that if the ring is also modular then it is a projectivity-UFD (definition of the latter recalled below)     

On Automatic Rees Matrix Semigroups

We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU ¹ = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.     

Fuzzy Ideal Based Computational Approach for Group Decision Making Problems

In this paper, we present a computational method to fuzzy group decision making problems. A function that satisfies the properties of fuzzy ideal of semiring of positive integers is also investigated in the present paper and is used for idealizing the group preference matrix obtained by different decision makers. The proposed method appears in form of simple computational algorithms to idealize the group preference matrix and calculating total order of preference relation. Finally, the suitability of the proposed method is shown by taking an example of a human resource development (HRD) event, where it is used to select the best possible candidate by different decision makers.