### Group ideals in a semigroup of measures.

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The concept of strong spined product of semigroups is introduced. We first show that a semigroup S is a rpp-semigroup with left central idempotents if and only if S is a strong semilattice of left cancellative right stripes. Then, we show that such kind of semigroups can be described by the strong spined product of a C-rpp-semigroup and a right normal band. In particular, we show that a semigroup is a rpp-semigroup with left central idempotents if and only if it is a right bin.

In this paper, we prove that the class of P₂-Almost Distributive Lattices and Post Almost Distributive Lattices are equationally definable.

The concept of super hamiltonian semigroup is introduced. As a result, the structure theorems obtained by A. Cherubini and A. Varisco on quasi commutative semigroups and quasi hamiltonian semigroups respectively are extended to super hamiltonian semigroups.

Green's relations and their generalizations on semigroups are useful in studying regular semigroups and their generalizations. In this paper, we first give a brief survey of this topic. We then give some examples to illustrate some special properties of generalized Green's relations which are related to completely regular semigroups and abundant semigroups.

It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. We have recently extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper, we introduce the notions of Clifford semidomain and Clifford semifield. Some structure theorems for these semirings are obtained.

Let $S=\{{x}_{1},\cdots ,{x}_{n}\}$ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $({x}_{i},{x}_{j})$ (resp. the $e$-th power of the least common multiple $[{x}_{i},{x}_{j}]$) as the $(i,j)$-entry of the matrix by $\left({({x}_{i},{x}_{j})}^{e}\right)$ (resp. $\left({[{x}_{i},{x}_{j}]}^{e}\right))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $({x}_{i},{x}_{j})\in S$ (resp. $[{x}_{i},{x}_{j}]\in S$) for all $1\le i,j\le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that...

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