We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

中文翻译：

---

半环上的矩阵半群

我们研究了由半环上的下列矩阵半群族中的幂等性决定的性质 [公式：见正文]：全矩阵半群 [公式：见正文]、由上三角矩阵组成的半群 [公式：见正文]，以及由所有单位三角形矩阵组成的半群 [公式：见正文]。Il'in 已经证明（对于 [Formula: see text]）半群 [Formula: see text] 当且仅当 [Formula: see text] 是正则环。我们证明 [Formula: see text] 是正规的当且仅当 [Formula: see text] 且 [Formula: see text] 的乘法半群是正规的。丰富或喷泉（以前，弱丰富）的概念弱于规则，但也是根据幂等定义的，即每一类某些等价关系都必须包含一个幂等。[Formula: see text]、[Formula: see text] 和 [Formula: see text] 中的每一个都承认一个自然的反同构，允许我们根据列空间上的幂等矩阵的左作用来表征丰度和 Fountainity。在半环是精确的情况下，我们证明 [Formula: see text] 是丰富的当且仅当它是正则的。我们的主要兴趣在于 [公式：见文本] 是幂等半场的情况，我们的动机示例是热带半环 [公式：见文本]。我们证明了 [公式：见正文] 的某些子半群，包括经过充分研究的二元关系（霍尔关系、自反关系、单三角布尔矩阵）的一些推广，是喷泉。我们还考虑子半群 [Formula: see text] 和 [Formula: see text] 由 [Formula: see text] 和 [Formula: see text] 的矩阵组成，所有元素都在前导对角线非零之上。我们证明[公式：见文本]的幂等生成子半群是[公式：见文本]。此外，[Formula: see text] 和 [Formula: see text] 是 Fountain 半群的族，具有有趣和不寻常的性质。特别是，每个 [Formula: see text]-class 和 [Formula: see text]-class 都包含一个唯一的幂等性，其中 [Formula: see text] 和 [Formula: see text] 是用于定义 Fountainicity 的关系，但是幂等不形成半格。见正文] 和 [公式：见正文] 是具有有趣和不寻常性质的 Fountain 半群族。特别是，每个 [Formula: see text]-class 和 [Formula: see text]-class 都包含一个唯一的幂等性，其中 [Formula: see text] 和 [Formula: see text] 是用于定义 Fountainicity 的关系，但是幂等不形成半格。见正文] 和 [公式：见正文] 是具有有趣和不寻常性质的 Fountain 半群族。特别是，每个 [Formula: see text]-class 和 [Formula: see text]-class 都包含一个唯一的幂等性，其中 [Formula: see text] 和 [Formula: see text] 是用于定义 Fountainicity 的关系，但是幂等不形成半格。