Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Monoids

Grzegorz Bancerek
Polish Academy of Sciences, Institute of Mathematics, Warsaw

### Summary.

The goal of the article is to define the concept of monoid. In the preliminary section we introduce the notion of some properties of binary operations. The second section is concerning with structures with a set and a binary operation on this set: there is introduced the notion corresponding to the notion of some properties of binary operations and there are shown some useful clusters. Next, we are concerning with the structure with a set, a binary operation on the set and with an element of the set. Such a structure is called monoid iff the operation is associative and the element is a unity of the operation. In the fourth section the concept of subsystems of monoid (group) is introduced. Subsystems are submonoids (subgroups) or other parts of monoid (group) with are closed w.r.t. the operation. There are presented facts on inheritness of some properties by subsystems. Finally, there are constructed the examples of groups and monoids: the group $\rangle{\Bbb R},+\langle$ of real numbers with addition, the group ${\Bbb Z}^+$ of integers as the subsystem of the group $\rangle{\Bbb R},+\langle$, the semigroup $\rangle{\Bbb N},+\langle$ of natural numbers as the subsystem of ${\Bbb Z}^+$, and the monoid $\rangle{\Bbb N},+,0\langle$ of natural numbers with addition and zero as monoidal extension of the semigroup $\rangle{\Bbb N},+\langle$. The semigroups of real and natural numbers with multiplication are also introduced. The monoid of finite sequences over some set with concatenation as binary operation and with empty sequence as neutral element is defined in sixth section. Last section deals with monoids with the composition of functions as the operation, i.e. with the monoid of partial and total functions and the monoid of permutations.

#### MML Identifier: MONOID_0

The terminology and notation used in this paper have been introduced in the following articles [16] [7] [21] [18] [10] [17] [1] [22] [8] [4] [2] [23] [6] [5] [3] [9] [19] [11] [12] [15] [14] [20] [13]

#### Contents (PDF format)

1. Binary operations preliminary
2. Semigroups
3. Monoids
4. Subsystems
5. The examples of monoids of numbers
6. The monoid of finite sequences over the set
7. Monoids of mappings

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