# Category Algebras and States on Categories

## Abstract

**:**

## 1. Introduction

**Notation**

**1.**

## 2. Category Algebras

**Definition**

**1**

- 1.
- R is a commutative monoid with respect to addition with the unit 0,
- 2.
- R is a monoid with respect to multiplication with the unit 1,
- 3.
- ${r}^{\u2033}({r}^{\prime}+r)={r}^{\u2033}{r}^{\prime}+{r}^{\u2033}r,\phantom{\rule{0.222222em}{0ex}}({r}^{\u2033}+{r}^{\prime})r={r}^{\u2033}r+{r}^{\prime}r$ holds for any $r,{r}^{\prime},{r}^{\u2033}\in R$ (Distributive law),
- 4.
- $0r=0,\phantom{\rule{0.222222em}{0ex}}r0=0$ holds for any $r\in R$ (Absorption law).

**Definition**

**2**

- 1.
- $r({m}^{\prime}+m)=r{m}^{\prime}+rm,\phantom{\rule{0.222222em}{0ex}}({r}^{\prime}+r)m={r}^{\prime}m+rm$ for any $m,{m}^{\prime}\in M$ and $r,{r}^{\prime}\in R$.
- 2.
- $0m=0,\phantom{\rule{0.222222em}{0ex}}r0=0$ for any $m\in M$ and $r\in R$.

**Definition**

**3**

**Definition**

**4**

**Remark**

**1.**

**Notation**

**2.**

**Definition**

**5**

**Theorem**

**1**

**Definition**

**6**

**Theorem**

**2**

**Theorem**

**3**

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

## 3. Example of Category Algebras

**Example**

**1**

**Example**

**2**

**Example**

**3**

**Example**

**4**

**Example**

**5**

**Example**

**6**

**Example**

**7**

## 4. States on Categories

**Definition**

**7**

**Definition**

**8**

**Theorem**

**5**

**Proof.**

**Theorem**

**6**

**Definition**

**9**

**Remark**

**3.**

**Definition**

**10**

**Definition**

**11**

**Theorem**

**7**

**Proof.**

**Definition**

**12**

**Definition**

**13**

**Remark**

**4.**

**Definition**

**14**

**Definition**

**15**

**Theorem**

**8**

**Proof.**

**Definition**

**16**

**Theorem**

**9**

**Proof.**

**Remark**

**5.**

**Theorem**

**10**

**Remark**

**6.**

**Theorem**

**11**

**Theorem**

**12**

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Saigo, H.
Category Algebras and States on Categories. *Symmetry* **2021**, *13*, 1172.
https://doi.org/10.3390/sym13071172

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Saigo H.
Category Algebras and States on Categories. *Symmetry*. 2021; 13(7):1172.
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**Chicago/Turabian Style**

Saigo, Hayato.
2021. "Category Algebras and States on Categories" *Symmetry* 13, no. 7: 1172.
https://doi.org/10.3390/sym13071172