Rees matrix seminearring

As a continuation of the work done in (R. Mukherjee (Pal), P. Pal and S. K. Sardar, On additively completely regular seminearrings, Commun. Algebra 45(12) (2017) 5111–5122), in this paper, our objective is to characterize left (right) completely simple seminearrings in terms of Rees Construction by generalizing the concept of Rees matrix semigroup (J. M. Howie, Fundamentals of Semigroup Theory (Clarendon Press, Oxford, 1995); M. Petrich and N. R. Reilly, Completely Regular Semigroups (Wiley, New York, 1999)) and that of Rees matrix semiring (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172). In Rees theorem, a completely simple semigroup is coordinatized in such a way that each element can be seen to be a triplet which gives this abstract structure a much more simpler look. In this paper, we have been able to construct a similar kind of coordinate structure of a restricted class of left (right) completely simple seminearrings taking impetus from (M. P. Grillet, Semirings with a completely simple additive semigroup, J. Austral. Math. Soc. 20(Ser. A) (1975) 257–267, Theorem [Formula: see text] and (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172, Theorem [Formula: see text]).

On exponent matrices of tiled orders

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent [Formula: see text]-matrices, and illustrate them finding all generating [Formula: see text]-exponent matrices. The generating [Formula: see text]-exponent matrices are found using a computer. We consider the Hasse diagram [Formula: see text] of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix [Formula: see text] there exists an oriented path in [Formula: see text] starting in some matrix unit and ending in [Formula: see text] which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix [Formula: see text] there exists a chain of non-negative exponent matrices [Formula: see text] such that [Formula: see text] is a [Formula: see text]-matrix, and each [Formula: see text] is obtained from [Formula: see text] by adding a [Formula: see text]-matrix.

Conjectures on additively divisible commutative semirings

We present a series of open questions about finitely generated commutative semirings with divisible additive semigroup. In this context we show that a finitely generated additively divisible commutative semiring is idempotent, provided that it is torsion. In the particular case of a one-generated additively divisible semiring without unit, such a semiring must contain an ideal of idempotent elements.

THE PRE-LIE OPERAD AS A DEFORMATION OF NAP

We introduce current-preserving operads, which are colored operads together with an additive semigroup structure on the set G of colors. This is the natural framework in which types of G-graded algebras can be dealt with. As an example of application, we deform the NAP operad into the pre-Lie operad in this context, where G is the additive semigroup of positive integers. More precisely, we give a family [Formula: see text] of current-preserving operads, depending on a scalar parameter λ, such that [Formula: see text], respectively, [Formula: see text], is the current-preserving operad governing graded NAP (respectively, pre-Lie) algebras.

Nonself-adjoint Semicrossed Products by Abelian Semigroups

Let S be the semigroup S = Σ⊕ki∈1 Si, where for each i∈ I, Si is a countable subsemigroup of the additive semigroup R+ containing 0. We consider representations of S as contractions {Ts} s∈S on a Hilbert space with the Nica-covariance property: T*s Tt = TtT*s whenever t^s = 0. We show that all such representations have a unique minimal isometric Nica-covariant dilation.This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of S on an operator algebra A by completely contractive endomorphisms. We conclude by calculating the C*-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

On Helly's principle for metric semigroup valued BV mappings to two real variables

We introduce a concept of metric space valued mappings of two variables with finite total variation and define a counterpart of the Hardy space. Then we establish the following Helly type selection principle for mappings of two variables: Let X be a metric space and a commutative additive semigroup whose metric is translation invariant. Then an infinite pointwise precompact family of X-valued mappings on the closed rectangle of the plane, which is of uniformly bounded total variation, contains a pointwise convergent sequence whose limit is a mapping with finite total variation.

Perforated Ordered K0-Groups

A simple C*-algebra is constructed for which theMurray-von Neumann equivalence classes of projections, with the usual addition—induced by addition of orthogonal projections—form the additive semigroup{0, 2, 3, . . .}.(This is a particularly simple instance of the phenomenon of perforation of the ordered K0-group, which has long been known in the commutative case—for instance, in the case of the four-sphere—and was recently observed by the second author in the case of a simple C*-algebra.)

THE COMPLEXITY OF COMPUTING PARTIAL SUMS OFF-LINE

Given an array A with n entries in an additive semigroup, and m intervals of the form Ii=[i,j], where 0<i<j≤n, we show that the computation of A[i]+⋯+A[j] for all Ii, requires Ω(n+mα(m,n)) semigroup additions. Here, α is the functional inverse of Ackermann's function. A matching upper bound has already been demonstrated.

A note on the multiplicative semigroup of models of Peano arithmetic

In a model of Peano arithmetic, the isomorphism type of the multiplicative semigroup uniquely determines the isomorphism type of the additive semigroup. In fact, for any prime p of , the function x ↦ px is an isomorphism of the additive semigroup with the multiplicative subsemigroup of powers of p. It was observed by Jensen and Ehrenfeucht [3] that for countable models of PA, the isomorphism type of the additive semigroup (or even the additive group) determines the isomorphism type of the multiplicative semigroup. (See Theorem 3 below.)In this note we will show that the countability restriction cannot be dropped. First, we show (as Theorem 2) that for uncountable models of PA the isomorphism type of the additive group never determines the isomorphism type of the multiplicative semigroup. Our main result is Theorem 5 in which we show that the isomorphism type of the additive semigroup need not determine the isomorphism type of the multiplicative semigroup, thereby improving upon Harnik [2], where Theorem 5 is proved under the assumption of ♢. For completeness, a sketch of the proof of the Jensen-Ehrenfeucht result is included.The history of this paper begins with Nadel's question, asked in 1981, whether the countability assumption can be eliminated in the Jensen-Ehrenfeucht theorem. Soon afterwards, Nadel obtained the strong counterexample of Theorem 2, which applied to the additive group rather than the additive semigroup. A result of Pabion [8] shows that such a strong result is not possible for the additive semigroup.